Approximate Molecular Orbital Methods by Research Signpost

Approximate Molecular Orbital Methods 2010 Editor

Edward A. Boudreaux Professor Emeritus, Department of Chemistry, University of New Orleans, New Orleans Louisiana, USA

Transworld Research Network, T.C. 37/661 (2), Fort P.O., Trivandrum-695 023 Kerala, India

Published by Transworld Research Network 2010; Rights Reserved Transworld Research Network T.C. 37/661(2), Fort P.O., Trivandrum-695 023, Kerala, India Editor Edward A. Boudreaux Managing Editor S.G. Pandalai Publication Manager A. Gayathri Transworld Research Network and the Editor assume no responsibility for the opinions and statements advanced by contributors ISBN: 978-81-7895-466-0

Preface Recent advances in the ability to perform large scale, high speed computations have made it possible to conduct quantum chemical calculations at the ab initio level on small, to moderate sized, polyatomic molecules. Similarly, Density Functional Theory (DFT), has evolved to the level of allowing bonding and structure calculations on rather sizable molecules, where in metal and/or non-metal atoms in closed or open shells are involved. Although the DFT method is utilized routinely by theorists and nontheorists alike, it is not without its problems. A recent article in Chemical & Engineering News, June 30, 2008, pp 34-36 provides a current up-date on the status of DFT methods. Basically, DFT does provide good geometries and equilibrium bond distances, but bond energies, charge transfer, electronic (as opposed to vibrational) spectra and associated properties are not reliably calculated. Paul von Rague’ Schleger, University of Georgia, USA, has stated that users of DFT may have been lulled into a false sense of accuracy; “the happy days of ‘black-box’ DFT calculations are over” (says Schleger). Efforts to correct the major problem, exchange correlation energy, has had only variable success. While it appears possible to taylor density functional which can be applied to specific systems, there is no functional which is universally applicable to all systems. The B3LYP functional seems to be the best for all general applications, but it too suffers shortcomings, particularly when applied to multiconfigurational open shells of heavy metal atoms. An over-view of the DFT problems have stated very succinctly by Treg Van Voores, MIT, “If you have 18 different functional you may get 18 answers, you can probably find one that will reproduce what you want it to say. You lose the rigor of being able to say, ‘this is wrong’.” In light of these kinds of comments from respected experts in the field, all that can be said is that DFT methods, while accurate in principle or no better that semi-empirical in practice. Obviously, approximate and tested semi-empirical molecular orbital methods are by no means removed from the picture at this point in time, hence the purpose of this book. The first part covers the basic theoretical foundation and applications of the SCMEH-MO method, which has been developed and tested over a period of some 40 plus years. It does not require any adjustable parameters or ‘modified functionals’, so to speak. A variety of characteristic examples are presented in which the reliability of the method is clearly vindicated. These applications involve both open and closed shell systems, many incorporating

heavy metals with strong spin-orbit coupling. Bond energies, electronic spectra, magnetic properties, etc., have been reliably calculated in all instances. The second part focuses exclusively on the application of the INDO/CIS method to super molecular systems comprised of 1500-2000 electrons. Application is made regarding liquid solvent effects on the electronic spectra of pyrimidine in water and beta-carotene in acetone and isoprene solvents. The solvent structure environment is generated via Monte Carlo simulations, followed by application of the INDO/CIS procedure. Agreement with experimental spectral data is exceptionally good. While their have been a variety of semi-empirical methods developed and applied over the years, it is not goal of this book to review them. The sole intent as presented here is to focus on specific methods developed and /or applied by the authors Edward A. Boudreaux

Contents

Chapter 1 Self consistent modified extended Huckel molecular orbital method (SCMEH-MO) Edward A. Boudreaux Chapter 2 Molecules in different environments: Solvatochromic effects using Monte Carlo simulation and semi-empirical quantum mechanical calculations Kaline Coutinho, Tertius L. Fonseca and Sylvio Canuto

Transworld Research Network 37/661 (2), Fort P.O. Trivandrum-695 023 Kerala, India

Approximate Molecular Orbital Methods, 2010: 1-61 ISBN: 978-81-7895-466-0 Editor: Edward A. Boudreaux

1. Self consistent modified extended Huckel molecular orbital method (SCMEH-MO) Edward A. Boudreaux Professor Emeritus, Department of Chemistry, University of New Orleans New Orleans, Louisiana, USA

I. Introduction Ever since its initial development, the self consistent modified extended Huckel molecular orbital method (SCMEH-MO) has demonstrated the capability of producing highly reliable results comparable to ab initio methods and even superior to these. The first application was to octahedral, hexacoordinated fluorocomplexes of some first row transition metal ions [1]. Subsequently, a number of examples differing in size and kind have been successfully treated by the SCMEH-MO method [2-12]. This particular discussion will focus on both previously published and unpublished data.

II. Theoretical foundation In the interest of conserving space, the complete theoretical development of the SCMEH-MO method shall not be presented here; thus the reader is referred to prior publications for these details [1,3]. The effective, one-electron Correspondence/Reprint request: Dr. Edward A. Boudreaux, Professor Emeritus, Department of Chemistry University of New Orleans, New Orleans, Louisiana, USA. E-mail: eaboudre@yahoo.com

Edward A. Boudreaux

Hamiltonian of a molecular cluster, HefMol is factored into its separable components as

H effmol =HT + HV + HEX + HSP + HSO + HLF

(1)

where HT is the kinetic energy operator for the electrons considered in all the atoms of the cluster, Hv is the potential energy operator, HEX is an external electrostatic potential operator to buffer the charge of the molecular complex ion, Hsp is an effective spin pairing energy operator, and Hso and HLF are spin orbit and ligand field operators, respectively, that may be included where appropriate. It has been shown that the diagonal elements of the Hamiltonian reduce to the following [6]: Hii = ε(q)I ((j) + ε(q-1)i – EAi + ∑k ∑ij (q/Rii)k

(2)

for which ε(q)I is the valance orbital ionization energy for the electron in the ith AO of an atom having the charge q; ε(q-1)i is the corresponding valence orbital electron affinity; EA.i is the average electron affinity of the k neighboring atoms associated with the ith AO; (q/Rii) are all the local electrostatic (Madelung) first order terms for the ith AO in the environment of k atoms having q charges. Similarly, the off diagonal elements are expressible as: Hii = Hji = (2-|Sij|) Sii [(Hii + Hjj)/2]

(3)

where Sij is the two-center overlap integral, chosen so that rational invariance is preserved. The remaining terms in Eq. (1) are given as follows: HEX is the external potential energy operator for the external environment of the molecular cluster having a net charge, Q, and is given by n

∑ (V j =1

) j = −(−Q / n) / R AB

where n is the number of atoms surrounding a particular atom and RAB is the inter-nuclear distance between atoms A and B. The last three terms, HSP, HSO and HLF are treated as perturbations on the one-electron MO eigenvalues. HSP is expressed as an average spin pairing energy for each MO according to the relation [la)] E SP =

1 Ci2 Di [< S ( S + 1) > i − S ( S + 1)] ∑ 2 i

(4)

Theoretical foundation and applications

where Ci is the coefficient for the ith AOs in the MO, Di the associated atomic spin pairing parameter (evaluated independently), <S(S+1)>I, the average effective spin in each AO having an electron configuration ℓx (x = number of electrons, which are the populations calculated in the MO routine). This is expressed as:

< S ( S + 1) >= [ Peff ( Peff + 2) / n − (

2λ + 1 ) Peff ( Peff − 1) / 2]i 4λ + 1

(5)

where Peff = P/(2S+1), the effective Lбwdin population and S is the net spin for the MO. The elements of Hso are taken as the effective sum of spin-orbit coupling constants, A., obtained from HF-SCF calculations for electrons, in particular, AO configurations of atoms having the appropriate charge. Thus, ESO is determined by: ESO =

∑C

2 i

λi ( q, P )

(6)

where the λ(q,P) are taken as the appropriate weighted averages of those calculated for atoms having integral atomic charges (q) and populations (P), respectively. Finally, the HLF elements are explicitly calculated, via the usual Taylor's expansion, for an electron in the ith AO surrounded by a specific geometric array of effective electrostatic charges. The AOs in the basis set are, in principle, not restricted to any specific type, but are usually chosen as single STOs with exponents ζ and effective principle quantum numbers, n'. These data are optimized from HF-SCF calculations of <r> and <r2> using the expressions:

[

−1 2 ζ = (n'+1/2) <r>-1 = (2n'+2)(2n'+1)(4 < r >

]

1/ 2

(7)

with integral n'. Cusachs has shown that these exponents do give an accurate representation of overlaps [13]. It has also been shown that the diagonal elements of Eq. (2) may be taken from atomic spectral data (as in usual Hϋckel type calculations) or directly from the atomic HF-SCF calculations, and fitted to the quadratic relation [7] Hii = Aqi2 + Bqi + C

(8)

The latter approach is now used almost exclusively, so as to eliminate any semblance of empiricism. Any of the required AO parameters may also be obtained from quasi-relativistic HF-SCF calculations, and these too have

Edward A. Boudreaux

been incorporated in a number of instances in calculations on platinum compounds [46, 8-10].

III. Applications and results A. MF6n complexes The first thorough test of the SCMEH-MO method was on MF6n -octahedral complex ions (M = Ti3+, Cr3+, Fe3+ and Ni2+) [lb]. We shall stress only the results for calculated electronic spectra and spin densities in this report. 1. Electron spectra Although numerous attempts have been made to correlate observed electronic spectra with assignments made from MO theory, the general conclusion made thus far is that it is difficult to obtain accurate results in the absence of a detailed calculation of both the ground and excited states involved. The main problem is that even if the geometries of the ground and excited states are essentially unchanged, the differences in spin properties of the two states are sufficient to account for discrepancies in results that are based merely on the ground state MO configuration. Only in those instances where the differences in spin interactions in the excited, compared to the ground, state are small or accidentally cancel each other, should close agreement between calculated and observed spectra be anticipated. With the present method, however, the concept of an average effective spin can be readily applied to both ground and excited states, and thus it should be possible to attain a reasonably accurate account of spectra for at least most observable electronic transitions. The only drawback is that this procedure will not account for possible changes in symmetries of excited states. In the latter instances completely separate calculations would be required. Although it is not the purpose of this paper to treat the calculation of spectra in any detail, the calculated results for both ligand field (10Dq) and charge-transfer (ligand to metal; 3th, ---> 3eg, t2u â&#x2020;&#x2019; 2t2g, t2uâ&#x2020;&#x2019; 3eg) bands of the MF6n- complexes considered here are given in Table 1. The very important effect of the "spin-pairing" energy correction is clearly demonstrated in that the data in column 4 of Table 1 bring the calculated values of 10Dq into nearly exact agreement with the observed ones. However, in the case of charge-transfer bands, the "spin-pairing" correction is only of the order <1 kK (1 kK = 1000 cm-1) and hence is of little consequence in these cases. From among the various likely choices for assignments there appears to be at least one that is in good agreement with the observed data.

Theoretical foundation and applications

Table 1. Electronic spectra of NIF6n- complexesa One electron 2t2g→3e g Uncorr.

Corb

10Dq

3t1u->2t 2g

t 2u→2t 2g

3t1u→3e g

t 2u→3e g

Πu→t 2g

TiF 6 3- De Laat c This work

15.37 15.97 15.85

15.49

Obs

15.3,

CrF6 3Fenske, et al.d

43.33

52.87

59.18

68.72

19.3f

47.7f

16.54 6.88(12.7)

Offenhartzhh This work

15.49

116.16

12.68

16.44

27.82

36.76

40.50

49.44 7.50f

16.10i,

Obs

15.20j) FeF6 3Fenske, et al.d

15.97

Offenhartzh This work

7.50(13.9) 10.08

13.64

Obs

29.44

37.13 37.0, 47.6f

4.04(7.25) 6.94

7.30

Obsd energies in kK (1000 cm-1). for spin pairing interactions. cRef [15]. dRef [16]. eRef [17] fRef [14 a,b]. gRef [18]. hRef [19,20]. iRef [21]. JRef [22]. bCorrected

27.04

6.09

Offenhartzh

aAll

19.35

13.35f

NiF6 4Moskowitz, et al.e This work

13.64

7.30

7.25g

5.81

11.26

12.75

18.20

Edward A. Boudreaux

There are further difficulties, however, in attempts to match calculated and observed charge-transfer spectra. The experimental data are questionable since in most cases wherein the MF6n- complexes are involved, assignments are made on the basis of incompletely resolved peaks [14 a,13]. Also, the observed bands are quite broad, spanning at least 10 kK or more, and suggest the likelihood of more than one transition, even though the assignments are always labeled L(π u ) --> M(t 2g ). Furthermore, even in the cases where it may be assumed that the assignments are essentially correct, the involved transitions are from ligand orbitals, which are non-bonding, to metal orbitals, which are at least partially bonding in character. Thus it is quite possible that the geometries of the excited states are sufficiently different from that of the ground state, thus accounting for any appreciable discrepancy between the calculated and observed data. It is of interest to point out that the results of the present calculations give a good match between calculated and observed bands for the complete spectra of all the MF6n- complexes treated here, if one is willing to accept different assignments from those reported in the literature. This will be treated in greater detail in a future communication.

2. Spin densities The extent to which electron spin is transferred from a metal atom onto the ligands has for a long time been accepted as direct experimental evidence for covalent bonding in MF6n- complexes. The basic equations for this interaction are well established and are derived from the hyperfine interaction of an electron on the metal atom with the nuclear moment of the ligand. The important relations are [23-24]: fs = 2SAs|A2s fσ - fπ = 2S(Aσ - Aπ)/A2p

(9)

where As and Aσ - Aπ are hyperfine interaction parameters obtained from experiment, A2s and A2p are analogous quantities for the fluoride ion, S is the total spin of the metal ion, and the f parameters are fractional spin densities [25]. The As parameter arises from the well-known Fermi contact interaction which is a function of |φS(0)|2, the square of the radial wave function of an s orbital at the fluorine nucleus, which gives the s-electron density at the nucleus; A. and A.,„ on the other hand, are functions of the operator 1/r3 for the fluorine 2p orbitals. Additional expressions relating spin densities to

Theoretical foundation and applications

molecular orbital coefficients have been derived by Shulman and Sugano [25], fs = Ceg / 3 f σ - fπ = Ceg/3 - Ct2g/4

(10)

where the Ceg and Ct2g are the coefficients for the highest occupied eg and t2g orbitals. It is important to point out that the evaluation of fS and fσ - fπ from observed data was made using the total spin, S, of the free metal ion. However, this should rather be the "true spin" of the metal in the MF6ncomplex which is correctly represented by the net effective spin obtained from the population analysis. Thus the observed values of fS and fσ - fπ reported in the literature should be corrected for net effective spin as demonstrated by LaMar in the publication [26]. Since the calculations of spin densities and hyperfine interaction constants provide a very sensitive test as to the accuracy of molecular wave functions, we have made such calculations using the present MO method. The results are summarized in Table 2. The calculated values of fS and fσ - fπ are found to be generally in excellent agreement with the "spin-corrected" experimental values. Similarly, the calculated values of AS and Aσ – Aπ obtained from the f parameters, together with the net effective spins for each complex, are in equally good agreement with experiment. The only case for which there is any major discrepancy is in the values of fS and AS for NiF64-. Structurally speaking, NiF64- unlike the other MF6n- complexes considered here, does not exist as an isolated entity. Actually the KNiF3 crystal does contain NiF64octahedral clusters, but these are linked through continuous -F-Ni-F-Ni-Fchains which give rise to an antiferromagnetic exchange interaction [27]. This interaction has been treated theoretically by Wachters, whose results show that as a consequence of this exchange the coefficient of the fluorine 2s orbital in the 3eg MO should be increased by a factor of 10, while that of the nickel 3d orbital remains essentially unchanged [28]. Thus, there must be a proportional decrease in the coefficient of the fluorine 2p6 orbital. The new values of these coefficients for the 2s and 2pσ orbitals in the 3eg MO are 0.02480 and 0.43300, respectively. From these data the new calculated values of the superhyperfine parameters are fS = 0.83, fσ – fπ = 5.82, AS= 33.225, and Aσ - Aπ = 8.602, all of which are in nearly exact agreement with experimental results.

Edward A. Boudreaux

Table 2. Superhyperfine Spin Density at F in Some MF6n- Complexesa. Calc, This work CrF63-

Obs. Reported

Calc. Correctedb

Other works

fs fσ fπ fσ - fπ As Aσ - Aπ

-2.192e 4.989 -4.989

-8.918

-0.255e -6.0 ± 1.0c

-4.8 ± 0.8

-4.9 ± 0.8d

-3.9 ± 0.6

-4.742e

-4.5 ± 1.0c -7.2 ± 1.2d

FeF63-

0.61 ± 0.08

0.4333

fσ

8.547

6.336e

fπ

5.570

1.196e

fσ - fπ

2.977

3.40 ± 1,0f

17.171

3 ± 1f

3.370

3 ± 1f

0.083g

0.538h

fσ

15.177g

2.86i, 0.455e

fπ

8.614g

-0.003e

fσ - fπ

6.563g

3.78h

4.487g

33.9h

34.48i

8.938

8.8h

8.10i

As Aσ - Aπ NiF64-

0.8 ± 0.1d,g

Aσ - Aπ

2.66 ± 0.76

5.140e

0.847

0.46i

5.954

2.86i, 0.458e

in %, A in 104 cm-1. for effective spin interaction, as discussed in the text. cRef. [29]. dRef. [30]. eRef. [31]. fRef [32]. gSee text for adjustments to these values. hRef. [18]. iRef. [28]. bCorrected

B. HCo(CO)4 and Co(CO)4 The second example chosen to demonstrate the utility of the SCMEHMO method is the data obtained for the Co(CO)4 radical and the molecule containing an explicit cobalt-hydrogen bond, HCo(CO)4.

Theoretical foundation and applications

SCMEH calculations on the HCo(CO)4 molecule were carried out utilizing the valence electrons of Co (3d, 4s and 4p basis), one of H (ls and ls, 2s, 2p bases) and ten for each CO ligand (2s, 2p basis). These fifty electrons were placed in lA1 through 8A1 and 1A2 and 1E through 8E molecular orbitals, consistent with the C3, point group symmetry of the molecule. The coordinate system adopted placed Co at the origin, H and CO(1) ligands along the Z axis and CO(2)-CO(4) in the equatorial plane. Table 3 contains the pertinent data for some of the highest occupied MOs accounting for fourteen of the total fifty electrons in the molecule, and the unoccupied virtual orbitals up to the lowest positive 13E*. A comparison of the present results with other calculations is presented in Table 4. Unfortunately there is not much detailed information available in the published literature with which to make systematic comparisons. The data of Fonnesbeck, et al. [33], for example, provide the calculated energy levels but no orbital populations and atomic charges. Grima et al. [34], on the other hand, Table 3. Calculated MO Data for HCo(CO)4. Percent AO Composition per M0a MO Occ. Ecalb

Co 3d

C0c

4σ

5σ

6σ*

Iπ

2π*

0.7

0.3

31.4

10.7

1.3

0.2 40.8

5.2

8.0

3.8

0.6 46.5

4.1

0.1 36.8

2.4 45.7

0.8

2.8

2.9

0.4 10.0

0.3

12.2 61.6

0.4

0.6 28.6

1.3

16.6 46.0

13E*

+3.49

10A1

-2.09

9A1

-3.78

12E

-6.39

11E

-7.44

10E

-7.78

0.9

-8.44

7.4

8A1

-13.51

0.6

-13.90

5.9

7A1

-16.08

6A1

-16.84

9.3

5.9

2.3

7.8

74.0

0.7

5A1

-17.05

2.1

24.4

6.3

3.3

62.9

1.0

-17.32

2.2

1.9

1.4

0.6

87.8

4A1

0.2 0.3

6.4

31.4

0.3

6.5 3.2

2.3

0.5

Relative percent AO character within each MO. All in units of eV with spin pairing correction included. cThe small contributions for 3σ have been omitted.

0.2

8.5

5.2

Edward A. Boudreaux

Table 4. Comparative MO Parameters for HCo(CO)4. Reference (Method)

E(Occupied MOs):

FHJa (restricted HF) eV

GCKb

(CNDO)

(ab initio)

This work (SCMEH) eV

(A1) -10.63

(A1)

(E)

(A1) -13.51

(E) -10.70

(E)

(A1)

(E) -13.90

(E) -14.40

(E)

(A1) -16.08

(A1) -17.23

(A1)

(A1) -16.84

(E) -17.55

(E)

(A1) -17.05

(A1) -17.73

(A1)

(A1)-17.32

7.54

6.79

8.55(8.51)c

0.43

0.49

0.29(0.31)

0.55

1.49

0.59(0.71)

1.29

1.23

0.92(1.04)

Total Population Co

3d 4s

4p H

1.18

0.06(-)

0.25(-)

Cax 2s

1.39

1.17

1.37(1.38)

2.36

2.60

2.29(2.27)

1.79

1.17

1.58(1.58)

4.49

4.47

4.64(4.65)

Ceq 2s

1.46

1.19

1.37(1.41)

2p Oeq 2p

2.33 4.51

2.64 4.48

2.26(2.24) 4.61(4.62)

+0.59

+0.23

-0.43(-0.52)

-0.31

-0.29

-0.23

-0.23(-0.04)

+0.25

+0.23

+0.34(+0.35)

Oax

-0.28

-0.18

-0.22(-0.23)

Ceq

+0.21

+0.17

+0.37(+0.35)

-0.30

-0.19

-0.19(-0.21)

Oax 2s 2p

Net Atomic Charges

Oeq aRef.

[33]. (34). cValues in parentheses are for Hls basis only. bRef.

Theoretical foundation and applications

present detailed populations and charges but no orbital energies. Furthermore, the publication [34] is the only one which mentions any details as to the choice of basis functions. Yet their choice of a single 4p function with orbital exponent of 0.25 appears to be a rather poor representation for a well behaved Gaussian orbital. The Fonnesbeck et al. calculations [33] give the HOMO as an A1 orbital at -10.63 eV. While our results are in agreement with this symmetry assignment, the calculated energy is -13.51 eV. With the exception of the CNDO calculation [34] the data presented in Table 4 are all in agreement as to the symmetry of the HOMO. There are considerable differences, however, regarding the AO character of the HOMO. While the Fonnesbeck et al. [33] results suggest this to be an orbital of primarily Co 4pZ, 3dZ2 and Hls character, the Grima et al. [34] ab initio data indicate that it is primarily Co 3d & Hls with smaller contributions from the Co 4s and 4p. The CNDO results also agree essentially with Fonnesbeck et al. [33]. However, the SCMEH results presented in Table 4 suggest that the HOMO is primarily Hls and CO ligand, with a small degree of Co 4pZ character and only very small 3dZ2 and 4s contribution. A separate SCMEH calculation on the Co(CO)4 radical, shows that the unpaired electron resides in the 8A1 orbital at an energy of -11.08 eV. The relative percent AO character of this orbital is 8.5 Co(3dZ2), 1.2 Co(4s), 12.4 Co(4pZ) and 77.9 CO(σ). Hence, the result obtained when H is bonded, namely.

1. UV spectra The UV spectrum of HCo(CO)4 has been measured by Sweany in an argon matrix [38]. There is observed an intense shoulder at 227 nm and a broad intense band about 187 nm. There is no evidence whatever for any bands at longer wavelengths. Both bands appear to be charge transfer in character. From the data given in Table 3, and upon considering spin pairing corrections in the ground vs. excited state, the allowed single E A1 and Ai* 4-- A1 transitions are calculated. Assuming there is no appreciable geometry change or differences in net relaxation between ground and excited states, the relationship derived is: h λ Eh ← Eℓ = E corr − Ecorr − E sp (ex) / 2(Λ + 1)

(11)

where h and ℓ refer to the higher and lower states respectively, E sp(ex) is the average spin pairing energy for the excited configuration calculated according to Eqs. (4) and (5), and (Л + 1) is the maximum multiplicity of the

Edward A. Boudreaux

MO 'spin orbital' (MO degeneracy/Л = 0/0, 2/1, 3/2). The third term in eq. (11) is calculated to be 0.53/4 eV for singlet transitions from the 8A1 to 9E, 10E, 11E and 12E level respectively. The only allowed A*1 <— A1 transitions involve the 9A1 and 10A1 levels appearing at -3.70 and -2.09 eV. But these are too high in energy by comparison to the upper limits of the currently observed spectrum. In C3 symmetry E ←E transitions are allowed in both Z and X, Y polarization. Hence it is conceivable that transitions from the 8E to 9E, 10E, 11E and 12E levels may be competing with the E ←A1 transitions. For these singlet transitions, the third term in Eq. (11) is calculated to be 1.132/16 eV. However, it is to be noted that the 8A and 8E levels differ by only 0.39 eV and are hence likely to be highly mixed upon excitation. Other considerations for mixing include the 9E, 10E and 10E, 11E levels, which differ by only 0.66 and 0.34 eV, respectively. As shown in Fig. 3 of reference [38], there are two bands in the observed spectrum at 5.46 and about 6.6 eV, respectively, but the actual position of the latter is less certain since it appears just at the lower wavelength cut-off limit of the spectrometer. However, it is observed to be significantly even more intense than the other intense, rather sharp band at 5.46 eV. Nonetheless, both bands are clearly contained within an envelope of intense absorption having its on-set at less than 300 nm. Both calculated and observed bands in the UV spectrum of HCo(CO)4 and Co(CO)4 are presented in Table 5. It is to be noted that all calculated bands appear well within the observed absorption envelope, and there are no bands calculated at lower energies. Thus the agreement with the observed spectrum is good in this respect. It may further be inferred from Table 3 and 5 that the orbital characteristics of the pertinent levels involved in the symmetry allowed transitions (i.e., E ←A1 and E ←E) are such, that even spin-forbidden triplet transitions should have comparably large oscillator strengths and thus be of relatively high intensity. If such is the case, then the lower energy observed band (which is less intense) might be 93E 4— 1(8A1, 8E), which according to Eq. (11), would have the calculated energy: (-8.44) + (13.51 + 13.90)/2 + (0.53 + 1.132)/8 (eV) = 5.47 eV. This would obviously be in excellent agreement with the observed value. However, confirmation of this proposal will have to await a further detailed experimental spectral investigation. Furthermore, the relative AO characteristics of the MOs indicate that a 9E←(8A1, 8E) transition would have the effect of displacing all the electron density from the strongly bonding Hls orbital. Such would not be the case if the 10E orbital were comparably involved in the photolysis mechanism. A similar effect would also occur with transitions to the higher 11E and 12E orbitals.

Theoretical foundation and applications

Table 5. UV Spectrum of HCo(CO)4 and Co(CO)4. A. HCo(CO)4 E(h←ℓ )

M0a

7.4Co(3d),3.8H(2p),88.9C0

5.39

∆Ec (obs.,eV)

91E←81A1 91E←81A1 5.46

92E←

5.47

5.60

Assignments

(≥1%)

(calc.,eV) 4.94

Percent AO Character

1 (8A1 ,8E)

10E 1.0Co(3d),1.0Co(4s,4p),31H(1s) 101E←81A1

11H(2s),1.3H(2p),54.7C0 6.05

101E←81E

5.94 6.39

11E 1.0Co(3d),1.0H(2p),98C0

111E←81A1 111E←81E

7.06 7.51

12E 12Co(3d),2.0H(2p),86C0

121E←81A1 121E←81E

--6.6 (est)

B. Co(CO)4 ∆E(obs.)d,e AE(calc.)d

Ar matrix

CO matrix

28,316

29,410

28.090

32,913

33,780

31,650

41,706

41,670

39,120

Assignment 102E←72A1 112E←72A1 122E←72A1

a The 9E, 10E and 11E levels differ by only 0.66 and 34 eV respectively. Hence, both energy and symmetry dictate that these should be heavily mixed in the excited state. b All must be either E ←A1 or E← E transitions, since A1 ←A1 are much too high in energy. cSee

Ref. [381]

dcm4

eData

from Ref. [39]. Two unresolved shoulders ca. 39,000-29,000 cm-1, not included here.

Thus it may well be that both bonds in the observed spectrum are photoactive. The observation that upon photolysis the ratio of CO to H loss is about 8:1 (neglecting the cage effect for hydrogen atom loss) can also be rationalized from these calculated results. As can be noted from Table 3,

Edward A. Boudreaux

those bonding MOs having a large degree of H character contain about twice as much CO character as well, and there are four times as many COs per mol as H. Furthermore, there are other transitions in the 6-7 eV range for which the amount of CO character in the vacant MOs is greater than the occupied, which can reasonably be expected to cause metal carbonyl bond weakening. Hence, a sizable CO to H loss ratio is not unexpected on the basis of this calculated electronic structure.

2. Photoelectron spectrum of HCo(CO)4 It has been clearly demonstrated by the work of Bacon and Zerner [36], Tse [37], Tossell [40], Bohm [41a,b], Larsson [42], Guest et al. [43], and Calabro and Lichtenberger [44], to cite a few among many, that the application of Koopman's theorem to the photoionization of transition metal complexes gives rise to enormous errors, owing to large relaxation effects accompanying the ionization of metal orbitals. In fact, Guest and co-workers have made the explicit comment that their ab initio SCF calculation of the HCo(CO)4 ground state gives an incorrect order (as far as correlation with the observed P.E. spectrum via Koopman's theorem is concerned) for those MOs which have been assigned Co-3d and Co-H bonding [43]. A ∆SCMEH calculation analogous to ∆SCF calculations has been carried out in the following manner. The energy of the HCo(C0)4+ ion is calculated for each occupied MO from which a hole has been created for the (0) → (+) orbital ionization. These calculations were made by considering the net atomic charges (q) calculated for the HCo(CO)4 unionized state to be increased to (q + 1) in the ionized state. The atomic inputs for the ion calculation were determined for the (q + 1) ionized atoms with relaxation included. The latter correction was computed from the relation given by Calabro and Lichtenberger [44]:

(

)[

E R = ∑ N n+,λ / n 2 S n+,λ − S n ,λ

]

(12)

n ,λ

where ER is the relaxation energy, Ni-n,1 is the number of electrons in the ionized atom having quantum numbers n and ℓ, and S+n,t and Sn1 are the net shielding factors for ionized and unionized atoms respectively. The results of this ∆SCMEH computational procedure are presented in Table 6. All occupied MOs in the -19 eV range were considered in the calculation. Since the unionized HOMO (8A1) appears at -13.51 eV there are no occupied MOs at energies higher than this. The only other occupied MOs which were not included are those that appear lower than -20 eV, for which

Theoretical foundation and applications

Table 6. ∆SCMEH Calculation for Ionization of HCo(CO)4. HCo(CO) 4 Ionized MO

Percent AO Character a

HCo(C0) 4 + Eb

Percent AO Character a

SCMEH b

82(3d),18(CO)

-23.55 97(3d),3(CO)

8.57(8.84) c

55(3d),45(CO)

-20.87 98(3d),2(CO)

9.19(7.37)

5A 1

52(3d),11(H,1s),37(CO)

-17.32 69.5(3d),9.6(4s),7(H,2p),11.9(CO)

9.92(4.81)

8A 1

6(4p),31(H,1s),63(CO)

-13.51 43(4p),13(H,2p),44(CO)

11.32(1.71)

6(3d),6.5(4p),5.4(H,2p),82.1(CO)

-13.90 4(3d),18(4p),75.7(H,2p),2.3(CO)

13.88(0.92)

7A 1

3.2(4s),2.3(H1ls),94.5(CO)

-16.08 1(4s),99(CO)

16.42(0.34)

6A 1

2.4(3d),0.5(4p),97.1(CO)

-16.84 100(CO)

17.19(0.35)

5A 1

2.4(3d),1(4p),0.5(H11s),96.1(CO)

-17.05 100(CO)

17.40(0.35)

4(3d),0.8(4p),95.2(CO)

-17.48 1.6(3d),98.4(CO)

2(3d),98(CO)

-17.76 100(CO)

18.46(0.70)

9.2(3d),90.8(CO)

-17.68 6(3d),94(CO)

18.48(0.80)

18.21(0.73)

Only those AOs contributing 1.0 percent are included. All in eV, including spin-pa iring interaction, etc. c Only the relaxation contribution b

even relaxation corrections would not bring them into the 8-19 eV range of the observed photoelectron spectrum of HCo(CO)4 [45]. A comparison of the calculated and observed photoelectron spectrum is presented in Table 7.

C. Magnetic parameters for the Co(CO)4 radical The basic definitions of magnetic hyperfine A and g tensors may be found in any standard text [46]. In an axially symmetric geometry the anisotropic components of the A tensor for 59Co in Co(CO)4 (C3V-z totally symmetric axis) are given as A║ (59Co) = AFC + Azz(DIP)1 + Azz(DIP)2

(13)

A┴ (59Co) = AFC + AXX(DIP)1 + Axx(DIP)2 where only the diagonal components of Aqq(q = x, y, z) are non-vanishing and Axx= Ayy. AFC is the Fermi contact term given by

Edward A. Boudreaux

Table 7. ASCMEH Calculated vs. Observed Photoelectron Spectrum of HCo(CO)4. ∆E(+)<-(0)

P.E.S. (obs.)

Band Charactera predicted obs.

Primary AO Composition

8A1

H and CO

11.32

11.5

s-ns (H,CO)

s-ns (CO-H)

5A1

Co and CO

9.92

9.90

b-ws (Co,CO)

b-ws (Co)

Co and CO

9.19

8.57 (ave=8.88)b

8.90

b-ws (Co)

7A1

16.42

6A1

17.19

5A1

17(tail)

b-ns (CO)

CO(Co)

13.88

b-ss (CO,Co)

b-fs (Co-C)

18.21

18.48

b-ns (CO)

? (CO)

(calc.)

17.40 (ave=17.00)b

18.46 (ave=18.38)b

13.8(onset)

18.2

as,

sharp; b, relatively broad; ns, no splitting; ws, with splitting; ss, some splitting; fs, fine structure. bOrbital characteristics suggest these excited states should be thoroughly mixed.

AFC =

8π g e g n β e β n ∑ C ki2 |ψ(0)ki|2 (Pki) 3

(14)

where geβe and gnβn are electron and nuclear 'g factors' and magnetic moments respectively (βe is given a negative sign); Cki are the Lowdin AO coefficients for each orbital in the molecular orbital containing the unpaired electron; ψ(0)ki is the electron density per electron for the ith orbital at the nucleus of the kth atom, and Pki the Lowdin populations per orbital per atom. A(DIP)1 and A(DIP)2 are the first and second order dipolar contributions given by

Theoretical foundation and applications

A(DIP) 1 = g e g n β e β n

∑ C (α 2 k ,i

k ,i

A(DIP)2= − 2 g e g n β e β ~

)

qq ' m , mλ k ,i

∑∑ C (α 2 ki

< r −3 > k ,i ( Pk ,i )

)

qq ' m , mλ ki

r −3

(15a)

( Pki ) x

ψ 0 | ς λ Lq | ψ n ψ n | Lq | ψ 0 En − E0

(15b)

qq ' where, in addition to the terms already defined, α m , mλ is the orbital angular

factor for the alignment of the electron spin according to a (3 cos 2 θ ℓ - 1) dependence, relative to spin (m) and orbital (mℓ) coordinates, q and q', respectively; <r-3> is the one-electron orbital expectation value for kth atom in the calculated charge state; ζℓ is the one-electron spin-orbital coupling constant for each of the k atoms comprising the LCAO-MO containing the

unpaired electron, and Lq is the qth (x, y, z) component of the orbital angular momentum connecting the ground, ψ0., and excited, ψn., states, having the energie E0 and En respectively. It is to be noted that in the present case, the effective radial distribution of the cobalt 3d, 4s, and 4p orbitals is 3.09 au (weighted average) which is to be compared with the cobalt-ligand bond distance of 3.33 au (1 au = 0.53 A). Since the radial distributions of the carbon and oxygen 2p orbitals are about 1.5 and 1.3 au respectively, the unpaired electron in the Co(CO)4 radical has its spin density well localized onto the CO ligands. Hence it is appropriate to evaluate the tensors of Eqs. (14), (15a) and (15b) by summing over i orbitals of the k atoms comprising the LCAO-MO in which the unpaired electron resides. The associated <r-3> terms are evaluated as projected onto the Co nucleus, which of course for the CO ligands, turn out numerically to be essentially the same as those for free C and O atoms appropriately adjusted for their respective net charges in the molecule. The data for AFC of 59Co in Co(CO)4 obtained from the SCMEH calculation of AFC are presented in Table 8. Since only s orbitals give non-vanishing contributions to |ψ(0)2 for a non-relativistic basis, these were the only contributions to AFc considered according to Eq. (14). Similarly, the calculated A(DIP)1 and A(DIP)2 terms of 59Co, are presented in Table 8. The ligand hyperfine 13C A tensors may be written as A ║ (13 C) = AFC + 2[AL(DIP)0 + AL(DIP)1 + AL(DIP)2]

(16a)

A ┴ (13 C) = AF.C. -[AL(DIP)0 + AL(DIP)1 +AL(DIP)2]

(16b)

Edward A. Boudreaux

Table 8. Magnetic A and g Tensors for Co(CO)4. Co

13C

Calc. (10-10au)

Obs.a (10-19au)

-145.05

-141

A(DIP)1 Azz

409.56

AXX AYY

-163.73 -163.49

A(DIP)2 Azz

2.86

AXX

59.29 -60.95

AYY

261.6 -248.4

264 ± 5b 250 -± 5b

A║

-112.8

A┴

112.8

119 + 9b 109 ± 5b

A║ A┴

g║ g┴

2.0092 2.1356

2.007 ± 0.010 2.128 ± 0.010

Data from Ref. [35]. Only absolute values are reported.

where the zeroth order ligand dipolar term is AL (DIP)0 = gegN βeβN [ 1/ < rMO >3eff ]

(17)

with <rMO>eff is taken as the weighted average of all ith AO's over k atoms in the MO containing the unpaired electron. All other terms in (16a) and (16b) are as defined in (15a) and (15b), but referred to the carbon rather than the cobalt nucleus. The g tensors are affected only by spin-orbit coupling and the extent to which the orbital angular momentum mixes ground and excited states via

Theoretical foundation and applications

second order perturbation. The appropriate equations for the axially symmetric case are

[

~ ~ g║ = ge + 2 λeff ∑ ψ 0 | Lz | ψ n ψ n | Lz | ψ 0 n≠0

g┴ = ge + 2 λeff x

∑[ψ n≠0

](E

− E0 )

−1

~ ~ ~ ~ | Lx | ψ n ψ n | Lx | ψ 0 + ψ 0 | L y | ψ n ψ n | L y | ψ 0

(18a)

](E

−E0

)

−1

(18b)

where λeff is the spin orbital coupling constant for all the electrons in a specific configuration, which is taken as ζ(Pki), i.e. the one-electron spinorbital coupling constant for the atom in the specific charge state in the molecule, times the Lőwdin orbital population. The other terms bear the same significance as those defined in Eqs. (15a) and (15b). The final results of the SCMEH calculation of A and g tensors for Co(CO)4 are presented in Table 8 together with the observed values reported by Ozin, et al. [35].

D. Calculation of bond energies It is usually difficult to infer an adequate description of bond energy from only ground state MO calculations, unless the complete potential energy curve is calculated. However, there are some features unique to the SCMEH method which allows ready access to a reasonable estimate of bond energies, from the ground state equilibrium calculation. The calculated SCMEH eigenvalues provide a manifold of levels, both occupied and unoccupied virtuals, which fall into the negative energy region of the total molecular potential well. The critical threshold for the bond breaking process should be associated with the primary bonding MO levels, but these will be augmented via perturbations resulting from excited interactions with the appropriate virtual MOs lying in the negative region of the potential well. The primary valence MOs can be defined as those bonding MOs which contain an appreciable ad-mixture of AO character from both of the bonding atoms, and are confined to only those MOs whose energies do not exceed the lowest ionization threshold of the molecule. The average weighted bonding contributions from these primary valence MOs are accessed from the square of the LCAO-MO coefficients, ci2 . It is defined that ci2 = 0.50 for a given AO on either one of the atoms in the bonded pair is the criterion for maximum bonding. Thus each of the calculated ci2 for each AO in a primary valence

Edward A. Boudreaux

MO is weighted ci2 /0.50 = ciw2 from which the c2iwt = ∑ ciw , is obtained. 2

2 iw

Hence the ratios c / c

2 iwt

provide the fractional weights by which the energy

contribution of each primary valence MO must be multiplied. These are summed to give the total weighted energy. The same weighted factors are also applied to the number of electrons weighting each primary valence MO, and these are summed to provide the total weighted number of bonding electrons, é. From these data the weighted total energy per electron, E / e , is calculated. The net interatomic overlap population (I0P) is output as a direct result of the Lőwdin orthogonalization, and may be described as the average number of bonding electrons per bond type (#e/B.T.). Thus the product of E / e with the IOP(#e/b.T.) and the number of bond types, (B.T.), i.e. the MO degeneracy per bonding pair of electrons, should provide the first part of the bonding energy relation. The second part is derived from the average excitation energy for primary valence electrons promoted to vacant virtual MOs. This may be ρ expressed as ∆E ex / n where ∆E ex is the average energy difference for N average electrons in symmetry allowed electronic transitions Ei virtual (unocc.) ←Ei primary (occ.). 2

In calculating ∆E ex / n the ciw / ciwt must be determined including those symmetry allowed transitions to excited states which involve only the specific bonded atoms in question. In the CO molecule, for example, these transitions include the occupied 4σ, 5σ and 1π to the unoccupied 2π* levels, for which E = 20.42, 6.92 and 10.91 eV respectively. When each of these transition energies is multiplied by the fraction of ground state character (i.e. the

2 ciw2 / ciwt in each case), the energy contributions appearing in the row

headed by ∆E ex / n in Table 9 are obtained. The sum of these contributions divided by the weighted number of bonding electrons gives the value of ∆E ex / n Hence the final expression derived for bond energy via the SCMEH method is

(

)

B.E. = E / e (IOP) (#e/B.T.) (# B.T.) - ∆E ex / n

(19)

Application of Eq. (19) to a variety of neutral inorganic molecules and complex ions, has yielded the bond energy data presented in Table 9. In those cases involving open shells, the stabilization via configuration interaction with core electrons, must also be included in computing ∆E ex / n .

Theoretical foundation and applications

Table 9. SCMEH Bond Energy Calculations. Molecule

IOP

E / e (e/B.T.) #B.T.

∆E ex / n (eV) B.E.(calc)a B.E.(obs)a

Ref.

C12

5.57

0.352

2.00

1.25

2.67

2.49

[46]

HC1

7.00

0.637

1.00

4.46

4.45

[46]

8.37

1.20

1.19

0.81

11.14

11.23

[47]

8.55

0.691

1.10

-0.27

6.77

XeF4

3.20

0.861

2.00

4.12

1.39

CuF2(g)

5.36

0.435

1.75

4.08

CuCI42-

7.76

0.465

1.54

-1.05

6.61

6.66

[48]

FeC142-

4.64

0.734

1.80

-0.22

6.35

6.25

[48]

4.46

0.488

2.00

0.22

4.13

(bond)CoC0(eq)

10.25 0.134

2.00

1.30

1.44

(bond)CoC0(ax)

10.25

0.163

2.00

1.30

1.75

1.55

[50]

-1.12

(ave=1.60) 2.42

2.5

[51]

PtC1

6.56-7.04 [46] 1.36

[46]

3.43-4.22 [46]

3.96-4.39 [49]

42-

HCo(CO)4

(bond)Co-H

2.095

0.156

1.00

a. All values in electron volts

It is notable that in Table 9, the calculated bond energies are in excellent agreement with the observed reports in all cases. There were no calculations attempted for cases other than the ones reported here.

E. Platinum complexes and the relativistic effect MO calculations involving very heavy metals such as Pt are not very meaningful unless relativistic and spin-orbit effects are accounted for. A pseudo-relativistic approximation is derived by assuming a summation over a "frozen spin" distribution within the AOs, thus arriving at "average spin, oneelectron" orbitals having the compositional format ∑tiℓi where ti is a proportionality factor based on the multiplicity of mj components appropriate to a specific spin-orbital type, having the designation j=(ℓ±s).

Edward A. Boudreaux

Thus non pseudo-relativistic s, p and d orbitals become described as: s’ = s1/2 p’ = 1/3 p1/2 + 2/3 p3/2 d’ = 2/5 d3/2 + 3/5 d5/2

(20)

in the pseudorelativistic format. The wave functions for the relativistic orbitals listed in Eq. (20) should strictly be solutions utilizing the full relativistic Hamiltonian, carried out to the Hartree Fock limit. Alternatively, one can utilize a quasi-relativistic approach [52], in which the mass-velocity and Darwin terms of the Pauli Hamiltonian are added to the HF differential equations for the atomic orbitals, resulting in the following relation,

Fˆ P nℓ (r) = e nℓ P nℓ (r)

(21)

where Fˆ is the Fock operator, Pnl(r) the radial wave function for the AO and the enℓ the associated spin function. The set of equations having the format of Eq. (21) are solved iteratively for both orbital energies and radial wave functions. Thus the most crucial direct and indirect relativistic effects are incorporated into the one-electron orbital function, so convenient for LCAO-MO calculations. This is not unlike the ab initio model-core and model-potential calculations, utilized for heavy atoms [53]. In molecules the orbital irreducible representations of the normal point group symmetry must transform into the corresponding irreducible representation of the appropriate double group. The irreducible representations and correlated spinors (ie., spin orbitals) for the tetragonal point group are given in Table 10. 1. Some simple Pt(II) complexes Both non-relativistic (NR) and pseudo-relativistic (PR) SCMEH-MO calculations have been carried out on the Pt(II) complexes: (I) PtC142-, (II) cisPt(NH3)2C12, (III) trans-Pt(NH3)2C12, (IV) Pt(NH3)42+ and (V) Pt(CN)42[4]. The results are presented in Table 11, as they relate exclusively to HOMO/LUMO characteristics of cases (I) through (IV). Diagrams correlating the Pt 5s 5p 5d 6s 6p AO characteristics are presented for NR-SCMEH results in Fig. 1. Similar diagrams correlating only the 5d orbitals are presented for PR-SCMEH results in Fig.2. An examination of Table 11.

Theoretical foundation and applications

Table 10. Orbital Representations and Spinors for D4h Point Group.

Edward A. Boudreaux

Figure 1. Non-relativistic SCMEH Results for some Pt(II) complexes (Pt basis: 5s 5p 5d 6s 6p). Abcissa (I): PtC142-; (II) cis-Pt(NH3)2C12, (III) trans-Pt(NH3)2C12; (W) Pt(NH3)42-; ordinate-energy (eV). Used by permission, Ref. [4].

Theoretical foundation and applications

Table 11. NR and PR-SCMEH Results for Complexes (I) Through (IV).

Figure 2. Pseudo-relativistic SCMEH. Results for some Pt(II) complexes (Pt basis: 5ar 2 5p 1/2 5p 3p 2 5d 3 p 2 5d 6Av 2 6p=). Abcissa (I): PtC142 - ; (II) cisPt(NH3)2C12, (III) trans-Pt(NH3)2C12; (IV) Pt(NH3)42-; ordinate-energy (eV). Used by permission, Ref. [4].

Edward A. Boudreaux

an Figs. 1 and 2 indicate some substantial fluctuations in bonding characteristics upon changing ligands. The data for both cis and transPt(NH3)2C12, for example, do in no way represent some average of the results for PtC142- and Pt(NH3)42+. The PR data are appreciably different from the NR both in orbital type and composition (not to mention HOMO/LUMO energy differences). It is particularly interesting to note that in the PR results, the HOMO of cis-Pt(NH3)2C12 contains some 5d and 6s character, but the trans isomer's HOMO is mostly ligand character with a very small amount of 6p, 5p Pt character. Perhaps this is significant to the fact that the cis isomer is an effective anticancer drug but the trans isomer is not.

2. Pt(II) - DNA interaction It has been well documented that cis-Pt(NH3)2C12 (DDP) is an effective anticancer drug, which binds primarily to the N7 position of the guanine nucleotide of DNA [54] (see Fig. 3, Case I). It was initially proposed that the findings of Pt induced a mispairing in the DNA. This required a breaking of the N1-H2 bond in guanine, which is attached to cytosine via hydrogen bonding. Thus SCMEH-MO calculations were carried out on: guanine (Case I), guanine-cytosine pair (Case II), DDP-N7 guanine (out of plane), and DDP-N7 guanine (in plane) (see Fig. 3, note Case II is not shown since the focus is on Pt binding and its influence in guanine). Note that the platinum moiety is regarded as Pt(NH3)2(H20) bound to N7, since there is ample evidence showing that the Cl is hydrolyzed in the reaction process [55]. The question regarding the effect of Pt-N7 binding on the N1-H1 bond of guanine is reflected in the Lowdin overlap population densities (LOPD) presented in Table 12. These are the sum of products of off-diagonal Lowdin eigenvectors between atom pairs, and may have either positive or negative values. But if the absolute values are considered, then fractional values which approach unity reflect a higher degree of covalent bonding, while those values further from unity are indicative of enhanced bond polarity. It is to be noted in Table 12 that the N1-H1 LOPD is lower for Case II than Case I, thus reflecting a weakening of the guanine N1-H1 bond upon base pairing with cytosine. Hence the influence of guanine (N1-H1) cytosine hydrogen bonding is clearly reflected. However, when Pt is bound to N2, there is no significant change in the N1-H1 LOPD (except for a slight reduction in Case IV). Thus these calculations do not support the belief that the N1-H1 guanine bond is affected by Pt-N7 bonding.

Theoretical foundation and applications

Figure 3. CASE I = Guanine, CASE III = DDP = N7 Guanine (out of plane), CASE IV = DDP â&#x20AC;&#x201C; N7 Guanine (in plane). Used by permission, Ref. [6]. Table 12. Lowdin Overlap Population Densities (LOPD). Bond

Pt-NH3 Pt-N7

III

0.12591 0.15207

0.13150 0.14616

Pt-OH2

0.02558

C6-N1 N1-H1

0.78117 0.57991

0.80860 0.48372

0.88424 0.61014

0.89505 0.52238

3. Electronic structure, bonding and spectrum of Pt2(P2O5H2)44- ie. Pt2(POP)4The general structural features of this molecular ion are shown in Fig. 4. QR-SCMCH calculations were carried out on both the ground and excited state structures of the complex, for which the Pt-Pt bond distances are 2.925 in the ground and 2.71 A in the excited states, respectively [56-59]. The total number of atoms in the cluster is actually 38 (2 Pt, 8 P, 20 O and 8 H). However, program limitations did not allow for calculations that exceed (m1 x m1) matrices with any array greater than (90 x 90). For these reasons the terminal oxygens and hydrogens were averaged as single atoms having an effective 2s overlap matched 2s'eff, orbital. The data required to accomplish this were generated from independent calculations on the P205H22 ion. This derived orbital, O'T(2s'eff) = 0.333 (H1s) + 0.2210(O2s) + 0.4460(O2p) for which the overlap matched STO orbital exponent is = 4.02321.

Edward A. Boudreaux

Figure 4. Coordinate system for Pt2(P2O5H2)44-. Used by permission, Ref. [8].

In the non-relativistic formulation the spin-pairing correction is given by Eqs. (4) and (5), by contrast the quasi-relativistic format is given as follows: ⎡P ⎤

⎛ j + 1/ 2 ⎞

⎞ eff ⎛ 2 E SP = − 2 ∑ Ci D'i ⎢ 2 ⎥⎜ j + 2 ⎟ − ⎜⎜ 4 j ⎟⎟ Peff (Peff − 1) − J (J + 1) ] ⎠ ⎝ i ⎠ ⎦⎝ ⎣ 1

(22)

where D'i is the relativistic Di, the average total spin-orbital quantum number, j , replaces ℓ, J the average total spin-orbit coupling, replaces S in Eqs. (4) and (5). The calculated quasi-relativistic one-electron MO eigenvalues and their respective symmetry assignments (C4h) for Pt2(POP)44- are given in Table 13 for the ground state (R(Pt-Pt) = 2.925 A). In each case the 136 electrons affiliated with the bonding (2Pt = 36: 8P = 40; 4Ob = 24; 16O’T = 32, plus 4 for the net charge) are distributed among 68 molecular orbitals. But Tables 13 and 14 do not include those MOs that are 95% or better pure ligand character; also those orbitals that comprise primarily the 5s, 5p core of Pt are not included. Assigning the orbital symmetry labels utilizes the full C4h rather than the local D4h symmetry. Since a number of the occupied MOs contain an

Theoretical foundation and applications

appreciable degree of OT character, attempts to describe the bonding in terms of D4h symmetry is not completely warranted. As shown in Table 13, the HOMO of the ground state is 7bg, with a 5au LUMO. In Table 14, it is shown that the HOMO is 9bg, but the LUMO is 5eg. In the latter case the unoccupied 5au level is 1.379 eV higher than the 9bg HOMO of the ground state. As far as electronic spectra are concerned, the pertinent levels involved in the strongest observed Î&#x201C;6-(Au) transitions must arise from the 4au and the deeper 3ag and 2ag levels. These are primarily 5dz2 in Pt character, but also contain an appreciable degree of O'T ligand character. The vacant levels to which symmetry allowed transitions can be made in the excited state are the 5ag and 5au levels, which are, respectively, ligand P 3s and Oâ&#x20AC;&#x2122;T 2seff, with some 6pz of Pt (5ag) and some 5dz2 (5au) character. Table 13. Non-Empirical QR-SCMEH MOs and Their One-Electron Energies of Pt2(POP)44. R(Pt-Pt) = 2.925 A, Ground State. MO

-E(eV)

6au

11.569

3au

20.842

5eu

11.574

5bu

20.932

5au

11.614

4bu

20.933

7bg (HOMO)

12.432

3bu

22.341

8bu

12.65

2eg

23.023

7bu

12.854

2bu

23.117

6bu

13.216

2bu

23.178

4au

16.965

2eu

23.235

4eg 4eu 5bg

17.215

3bg

23.540

17.592 18.548

2bg 1bg

25.789 29.446

6bu

18.552

1eu

30.217

3eu 3eg

18.709

1ag

32.686

18.932

1bu

35.049

3ag

18.953

1eg

35.634

4bg

19.416 19.504

1au

37.623

2ag

-E(eV)

Edward A. Boudreaux

Table 14. Non-Empirical QR-SCMEH MOs and Their One-Electron Energies of Pt2(POP)44- R(Pt-Pt) = 2.71 A, Excited State. MO

-E(eV)

5au 5ag 4eu

10.973 11.171 11.255

3ag 5bg 4bu

19.915 20.616 20.982

5eg HOMO 9bg

11.326

3au

21.029

7bu 6bu 8bg

12.352 12.587 12.620 12.969

3bu 2bu 2eg 2au

22.956 23.297 23.683 23.915

4au 4eg 3eu

16.811 16.912 17.891

2ag 4bg 3bg

24.162 24.581 26.848

7bg 5bu

18.601 18.747

2bg 1eu

29.244 30.049

2eu 3eg

19.000 19.098

1ag 1bu

32.531 35.237

4ag

19.213 19.821

1eg

35.775 37.636

6bg

1au

Details of the AO character, net atom charges, and Pt populations are presented in Table 15. From these results it is apparent that the MO levels associated with the electronic spectral bands are not merely Pt 5dz2 and 6pz in character, as previously presumed in more simplified approaches to the bonding in Pt2(POP)4- but do indeed involve a non-negligible amount of ligand character. a. Electronic spectral calculations. The intense transitions observed at 369 rim and 220 nm to be of singlet parentage [60] have Î&#x201C;6-(Au) symmetry assignment. As shown earlier, the only orbitals of appropriate symmetry which correspond to these energies are 2ag, 3ag, 4at, in the ground state and Sag, 5au in the excited state. Utilizing the one-electron eigenvalues in Tables 13 and 14, the appropriate calculated ligand field splitting (LFS), spin-orbit splitting (SOS) (for Pt), and the average spin-pairing energy, E SP , we calculated the total energies of the MO

Theoretical foundation and applications

Table 15. Characteristics of Key MOs in Ground and Excited State of Pt2(POP) 44â&#x2C6;&#x2019; Ground State, R(Pt-Pt) = 2.925 A. Total Pt Population: 5s=1.91, 5p=5.88, 5d=8.97, 6s=0.65,6p=1.44

Atom

5d z 2

4au

64.22

3.74

61.80

15.60

Charges

0.04

2.80

-0.866 1.03

0.01 19.41 a

0.40

9.38

Ob OT

3ag

0.24 0.04

0.01 10.93 a

OT 2ag

Pt P

AO Net

Per Cent AO Character 6s 6pz (Ligand)s

31.46

2.35

2.39

Ob OT

-0.247 -0.597

-3.63 7.75

-0.06 6.99 0.10 27.93 28.16 a

Excited State, R(Pt-Pt) = 2.71 A Per Cent AO Character Total Pt Population: MO

Atom

5au

Pt P

5ag

aThis

5s =1.90, 5p=5.88, 5d=8.99, 5dz2 6s 6pz 11.50

AO Net 6s=0.63, 6p=1.48 (Ligand)s

7.91

Charges

0.84

45.69

-0.882 1.05

0.00

1.10

-0.229

OT Pt

27.00 a 15.00

P Ob

3.51

37.81 a

is the OT(2seff) - HO sp mixed orbital. See text.

-0.605

0.00

34.40 9.28

Edward A. Boudreaux

ground and excited state configurations. In addition, it was also found that the relative MO eigenvalues are altered by several electron volts if the intermolecular 4 -(POP)4Pt2-Pt2(POP)44- i.e., Pt2-Pt2 - interaction is included. To account for this effort, we performed separate calculations on isolated monomeric Pt2P84+ and dimeric (P8Pt2)4+-(Pt2P8)4+ clusters, in which the intramolecular Pt-Pt distance in the latter case is 5.06 A [20]. Because of complications involved in conducting MO calculations on a molecular unit with non-integral charge, the atomic charges were fixed with Pt = -1.00 and P = 1.00. These are close enough to the converged values in the full Pt2(POP)44- complex so as not to significantly alter the interpretation of the intramolecular Pt2-Pt2 interaction. Thus the effect of the Pt2Pt2 interaction on the pertinent MO levels, relative to the isolated Pte situation, was assessed. The results are included together with the other data pertinent to the spectral calculations in Tables 16 and 17. As shown in Table 16, the calculated Γ6-( lAu), (4au)2→ (4au)1 (5ag)1 transition is 3.14 eV, which compares well with the observed value of 3.36 eV [61]. However, transitions originating from the deeper 3ag and 2ag levels must be rather thoroughly mixed, as neither of the calculated values presented in Table 17 for the isolated transitions is in close agreement with the observed. Yet the weighted average of these two transitions, Γ6-(1Au); (2ag)2 —> (2ag)1 (5au)1 and (3ag)2 --> (3ag)1 (5au)1, is 5.25 eV. This is to be compared with the observed values of 5.07 eV in the crystal and 5.64 eV in the solution spectra [57,60]. With the Pt2-Pt2 interaction removed, the band observed at 220 nm has a calculated energy of 5.89 eV. Thus it appears that the intermolecular interaction induces a red shift in this band. Further experimental support for this occurs because this band is shifted to about 4.71 eV in the [Pt2(POP)44-]X oligomer [61]. Table 16. Calculation of the 369-nm,Γ6-( 1Au). MO

(4au)2

E(eV)

E'(eV)

Total E(eV)

(no Pt2-Pt2

(with Pt2-Pt2

ELF (eV)

ESO (eV)

Esp (eV)

(with Pt2-Pt2

interaction)

-33.92

-39.78

interaction) 3.35

0.99

u 1

(4a )

-16.96

-19.74

1.83

0.48

(5ag)1

-11.17

-14.72

2.28

0.00

[(4au)1,(5ag)1] (net)

-28.13

-34.46

2.06

0.24

∆Eclc = 3.14; ∆Eobs. = 3.36 a Total

ligand field splitting. splitting for Pt only. c Total spin-pairing energy. d See Ref. [60]. b Spin-orbit

-0.62

-36.06

-0.76

-32.92

Theoretical foundation and applications

Table 17. Calculation of the 220-nm, Γ6-(1Au) Transition. MO

(3ag)2 (3ag)

E(eV) (no Pt2-Pt2

E'(eV)

Total E(eV)

(with Pt2-Pt2

ELF (eV)

ESO (eV)

Esp (eV)

(with Pt2-Pt2

interaction)

-37.90

-47.12

interaction)

2.92

1.90

-18.95

-23.82

0.92

0.55

(5au)1

-10.95

-14.52

0.12

0.08

(3ag)1, (5au)1 (net)

-29.92

-38.34

0.52

0.32

-1.36

43.64

1.84

-39.34

-2.81

47.55

-2.79

-41.37

AEcalc.d = 4.30

(2ag)2

-39.00

-48.22

2.50

0.98

-19.50

-24.52

0.55

0.17

(5au)1

-10.97

-14.52

0.12

0.08

(2ag)1, (5au)1

-30.4

-39.04

0.34

0.12

(2ag)

AEcalcd. = 6.18 aTotal

ligand field splitting. splitting for Pt only. cTotal spin-pairing energy. dWeighted average (relative to MO eigenvalues) from both 3ag and 2a, levels is 0.493 (4.30 eV) + 0.507 (6.18 eV) = 5.25 eV. The observed values are 5.07 eV (crystal), Ref. [571 and 5.64 (solution), Ref. [60]. bSpin-orbit

The Γ5-(3Au); (4au)2 →(4au)1 (5ag)1 transition has a calculated energy of 2.59 eV from the data in Table 16. The observed value is 2.74 eV [60]. b. Oscillator strength calculations. To further support the calculated spectra, we carried out oscillatory strength calculations. In a system as complex as this, it is not possible to proceed with these calculations by direct integrations over the appropriate molecular wave functions. The number of terms resulting from such an approach is too numerous to handle within present computer and program limitations. Thus transition dipoles for Zpolarized radiation were evaluated from a point-charge approximation given by:

∑ ( m / e) = ∑ ( q • R ) i

z i

(23)

where (m / e) i is the transition point-dipole per electron for each of the pertinent atoms summed over all the i atoms involved, q is the MO calculated net atomic charge on each of the i atoms, and Rz is the Zcomponent radius vector for the charge displacement associated with the

Edward A. Boudreaux

transition movement. The latter quantity is deduced from the observed structural data [59]. The oscillator strength, f, is given by the standard relation; f = 1.085 x -5 10 v x Peff2 a02 where v is the frequency of the transition in cm-1, Peff is the effective transition moment as defined subsequently, and a0 the Bohr radius. In this approximation the effective transition moment is given by Peff = ∑

{ 2C C (∆m / e) ' n

i 0

net

xr / st

}

(24)

Where (∆m / e) net is the net atom pair contributions of the transition pointdipoles, which are weighted with respect to r (the total dipole interactions of a given type), s (the number of electrons probable for the orbital transitions), and t (the number of different types of dipole interactions); C'n and C'0 are the net fractional AO contributions for all the orbitals involved on each of the atoms in the excited (n) and ground (0) states, respectively - these are evaluated from the square of the Lowdin eigenvectors in the LCAO-MO. The above relationships were applied with the Ob bridging atom of Pt2(POP)44- placed about the origin of the coordinate system - i.e., in the xy plane exactly at the midpoint of the observed Pt-Pt bond distance aligned in the Z-direction. The atomic point-charge dipole moments essential to the calculations are given in Table 18 for the Γ6− (1); (4au)2—> (4au)1(5ag)1 transition. The final Γ6−

Γ6−

Peff and f values for both the 369 nm, (1), and the 220 nm, (2), bands are given in Tables 19 and 20. The calculated f values are in good agreement with the observed f values. Table 18. Point-Dipole Moments of Pt2(POP) Rz(au)b

Pt P

-0.882 1.05

2.56 2.56

-2.26 2.69

-0.230 -0.605

0.00 4.08

0.00 -2.47

From MO results. From x-ray structural data, Ref. [60].

Qcalca

Atom

4− 4

( m / e)

Theoretical foundation and applications

Table 19. Transition Dipoles and Dipoles Strength for (4au)2 -> (4au)1(5ag)1 Transition (369 nm, obs.). Net Atom Fractions per MO

Atoms (n)

Cn'

(o)

C0’

( m / e)

2C n ' C 0 ' ( m / e) net ( r / st ) a

P Pt

Pt P

0.329 0.843

0.671 0.157

0.430 -0.430

√2(0.095) (8/2 x 12) = 0.045 √2 (-0.057) (8/2 x 12) = -0.027

0.107

0.893

-2.26

√2 (-0.216) (8/2 x 12) = -0.102

0.974

0.026

-2.26

√2 (0.057) (8/2 x 12) = 0.027

0.328

0.672

-4.73

√2(1.043) (16/2 x 12) = -0.983

0.436

0.564

4.73

√ 2 ( 0 . 0 2 7 ) (8/2 x 12) = 1.096

0.994

0.010

2.69

√2(0.027) (8/2 x 12) = 0.013

0.769

0.231

-2.69

√2(0.478) (8/2 x 12) = -0.225

0.661

0.339

0.220

√2(0.049) (16/2 x 12) = 0.047

0.931

0.069

0.220

√ 2 ( 0 . 0 1 4 ) (16/2 x 12) = 0.013

0.324

0.676

2.47

√2(0.541) (64/2 x 12) = 2.041

0.990

0.010

-2.47

√2(-0.025) (64/2 x 12) = -0.095

|Peff|| = 1.796 a. r = total dipole interactions of a given type; s = number of electrons probable for the orbital transition; t = number of different types of dipole interactions.

Table 20. Oscillator Strengths for the (4au)2→(4au)1(5ag)1, Γ6− (1) (398nm) and the

(2ag)2(2ag)1(5au)1,

Γ6

(2) (220nm) Transitions.

Transition

Γ6− (1) Γ6− (2)

Calc.(cm-1)

Peff2

f (calc,)

f (obs.)a

25,333

3.23

2.48x10-1

2.29x10-1

42,352

0.220

2.83x10

-2

~ 2x10-2 (shoulder)

a. See Ref. [62]

c. Pt-Pt bond energy calculation. It has been shown that the bond energy (B.E.) can be derived according to the present MO method from the relationship given by Eq. (19). This equation was shown to provide good

Edward A. Boudreaux

results for a variety of moderately simple molecules and complexes, but is not readily applicable to Pt 2 (POP) 44− There are many MO levels of various types involving the different degrees of Pt character, thus the Pt—Pt contribution alone cannot be essentially isolated from the Pt—ligand contributions. For these reasons the following was taken to arrive at the Pt—Pt bond energy. A separate MO calculation was conducted on the monomeric Pt (POP) 24− complex. The Pt contributions to each of the bonding MOs in both 4− Pt (POP) 24− and Pt (POP) 4 were deduced and the average energy per bonding MO was calculated. The difference between these average energies provides the average energy for binding two Pt atoms in the POP environment but uncompensated for spin pairing. Thus the net Pt-Pt bond energy is given as the sum of the average energy for binding two Pt atoms, plus E SP from Eq. (22). Both the Pt-Pt bond energy and the data utilized in determining it are contained in Table 21. The resulting B.E. = 8.05 kcal/mole seems reasonable for this system. This value is in agreement with the observation of Stein et al. [56] that the Pt-Pt force constant of 0.3 mdyn/angstrom, which reproduces the observed Raman frequency of the Pt-Pt mode in Pt2(POP)44-, follows the trend that a force constant of this magnitude correlates with a bond energy of the order of some 10 to 15 kcal/mole. d. Summary. It appears that the present study of the electronic structure and bonding in Pt2(POP) 44 − does provide an adequate explanation of the experimentally observed properties. These results suggest that the initial Table 21. Pt-Pt Bond Energy in Pt2(POP)4-q Complexes. Average Pt-Pt Spin Pairing Energy

Pt-Pt Binding Complex

E(Pt) All MOs E(Pt) per MO (eV)

(eV)

Pt(POP)42-

-378.28a

-8.223a

Pt2(POP)44-

-443.28b

-8.525

Energy, Eb

ESP

B.E.

(eV)

0.320

0.029c

0.349 (8.05 kcal/mol)

aPt-P

bonding only. plus Pt-Pt bonding, 39.5% total is Pt-Pt; the remaining 60.5% is Pt-P. cThe weighted average total contribution of all Pt-Pt bonding orbitals is reflected here. bPt-P

Theoretical foundation and applications

simplified MO proposal involving only Pt orbitals is not an adequate model for a quantitative rationalization of the complete electronic structure of Pt2(POP)44-. The calculated net Pt charge of -0.866 at the relatively short 2.92-A Pt-Pt bond distance and -0.882 at the even shorter 2.71-A bond distance in the excited state is consistent with both theoretical and experimental observations that there is very weak Pt-Pt bonding in this system. In fact, such large negative charge distributions do not allow for significant bonding between the pertinent atoms. The bond order, is given by the relation B.O. = ∑ NCi C j , where N is the number of electrons in the bonding MOs, and CiCi ij

are the coefficients of all the ith and jth AOs of the atoms in the LCAO-MO. Since Lőwdin orthogonalization is employed, the result is B.O. = ∑ ( Pop) ij ij

where (Pop)ij is the Lőwdin population in the ith and jth AOs of the pertinent bonding atoms in all the LCAO-MOs. For Pt-Pt bonding alone the result is B.O. = 1.00563 in the ground state and 1.24486 in the excited state. Thus, the excited state is about 11/4 times as strong or some 11% more bonding than the ground. In conclusion, it may be said that the instability of the Pt-Pt bond is a consequence of the large negative Pt charge which results in a low Pt-Pt bond energy and is not due to a low formal Pt-Pt bond order per se. Thus, the bulk of the stability of the Pt2(POP)44- cluster is provided primarily by the P-Ob-P bridge bonds.

F. Molecular laser dyes Here is an example for which the SCMEH method has been applied that is more organic than inorganic. This is one of the pyrromethene -BF2 laser dyes recently synthesized and characterized by Boyer and coworkers [63]. The 1,3,5,7,8-pentamethylpyrromethene-BF2 complex (PNIP-BF2) shown in Fig. 5 [64], was found to lase at a wavelength of 546 rim with 300% more efficiency than coumarin [63]. Thus it is of interest to examine the electronic structure of this molecule, particularly in so far as it relates to its electronic spectral behavior. Two calculations were carried out on PMP-BF2; one an ab initio Gaussian-88 (minimal basis set), and the other an SCMEH [65]. The comparative results from these two calculations are presented in Table 22, for the HOMO and LUMO levels. Subsequently, the lowest singlet (S-S) transition was calculated without any correlation or further optimization. Of course correlation is important in open shell systems, but not that critical in closed shells, so long as light atoms are involved. The SCMEH result predicts

Edward A. Boudreaux

Table 22. MO Data for PMPa-BF2. Ab Initio HOMO Atom C6

SCMEH

(Eb=-5.518)

LUMO (E=3.005)

pZ Coefficient

pz Coefficient

HOMO (E=-8.199) pz Coefficient

0.358

0.333

0.365

0.254

0.193

0.281

-0.195

-0.390

-0.207

-0.484

LUMO (E=-6.170) pz Coefficient 0.363 0.136 -0.370

-0.434 0.496

0.592

C10 C11

0.483

-0.101

0.439

-0.073

C12

-0.380

0.212

-0.393

C13

0.204 -0.245

0.156

-0.273

0.134

C14

-0.360

0.338

-0.372

0.362

aFor

-0.172

-0.257 -0.228

N1 N2

-0.170

ground state only. volts.

bElectron

H( 12)

Figure 5. ORTEP plot for 1,3,5,7,8-pentamethyl-pyrromethene-BF2. Used by permission, Ref. [64].

Theoretical foundation and applications

this S-S band to be at 518 nm, while the ab initio calculation gives 687 nm. The experimentally observed wavelength is 519 nm [63]. As can be seen from Table 22, the data provided by the SCMEH method is essentially equivalent to that of the ab initio. In fact the SCMEH calculation gives an excellent result for the singlet spectral band, while the ab initio does not. However, it is interesting that an approximate calculation of the oscillator strength, f, using point charge dipole relations (as described earlier) produced the value, f = 0.248. The value extracted from the observed spectrum is about 0.3. Both methods produced essentially equivalent results in this regard. The most significant features of these initial results are the comparisons of HOMO and LUMO levels. The HOMO contains carbon pz orbitals oriented perpendicular to the plane of the ring, and centered on ring-carbons 6,7,8,9 and methyl carbons 11,12,13,14, plus small contributions from associated hydrogen atoms. Interestingly enough, with the exception of C14, all of these ring carbons are positioned on that portion of the ring structure to the left and methyl substituents to the rear, of the BF2 respectively. It is curious that the electron density is increased on methyl C12 by a factor of two to three times that of any other atom in the molecule. The LUMO omits any contribution from C9 and includes the pz orbitals of C10, N1 and N2. Thus it appears that the lowest (S-S) transition transfers electron density from the alkyl substituents, across the it system onto the nitrogens. Consequently, any bonding that would more readily facilitate this intramolecular chargetransfer is desirable. This is apparently borne out by the fact that another compound in which the methyl group on the central ring has been replaced by an isopropyl substituent (see Fig. 5) does not show an observable spectral fluorescence [66].

G. The matter of correlation As stated earlier, correlation certainly does play a role in all systems, but it is particularly important in cases involving heavy atoms, for which unoccupied or partially occupied levels lie close in energy to the occupied levels. In order to attain an approximate assessment of the significance of correlation in SCMEH calculations, MO calculations were performed on a cluster appropriate to the structure of (NH4)2CuC14 [67]. This contains a Cu(II),3d9 open shell, in the environment of an axially distorted octahedral geometry. There is significant compression of the axial Cu-C1 bonds, as evidenced by the reported bond distances [67]: R(Cu-Cl)eq=2.34 and 2.30 A; R(Cu-Cl)ax = 2.79 A. In most copper chloride systems the axial bonds are more elongated and range from some 2.85-3.26 A [68]. This axial interaction is known

Edward A. Boudreaux

to exert a significant influence on the position of the Cu,3dz2 level, which apparently is shifted to higher energy as the axial bond strength increases. SCMEH calculations were made on a CuC14(NH4)4C12 cluster representative of all near-neighbor interactions in the crystal employing Cu3d,4s,4p, and Cl3s,3p basis sets. One calculation was conducted in the usual way, but a second one was made incorporating limited configuration interaction (CI) between MOs involving Cu orbitals only. A third calculation was also made utilizing the all-electron, ab initio, DMOL 2.0 routine [69]. This is a local density functional program which includes correlation out to the Hartree-Fock limit. The pertinent energy levels and their associated per cent AO compositions are given in Table 23. It is readily noted that the SCMEH results without CI show a series of occupied pure Cl, ligand levels, separating the Cu 3d levels, up to the HOMO containing one electron. However, the SCMEH-CI calculation shows that all the highest occupied levels are shifted upward in energy and contain Cu3d character. The pure Cl levels are pushed downward below the 3d. The same is also found to be the case for the DMOL 2.0 calculation, but the amount of 3d character is substantially greater. Table 24 contains the total orbital populations and atomic charges. Based on these data, it appears that the Cu-Cl bonds are some 1.7 times more ionic in the SCMEH-CI calculation than in DMOL 2.0. But the 3d orbital populations are essentially comparable in both calculations. Table 23. MO Results for CuC14(NH4)4C12 Cluster.

Theoretical foundation and applications

Table 24. Atomic Charges and Populations for CuC14(NH4)4C12. Charge Atoms

SCMEH-CI

0.97

Populations

DMOL 2.0 0.97

Orbital

SCMEH-CI

DMOL 2.0

9.75

9.54

4s 4p

0.273 0.020

0.490 --

C1((eq)x

-0.74

-0.45

5.74

5.45

Cl(eq)y

-0.72

-0.43

5.72

5.43

Cl(ax)z

-0.87

-0.76

5.87

5.76

(NH4 )

0.92

0.57

A calculation of the d-d electronic absorption spectrum was made via the SCMEH-CI routine. A comparable calculation with DMOL 2.0 was not possible because of the prohibitive amount of CPU time and expense involved. The pertinent optical transitions and comparative results are provided in Table 25. It is noted that agreement between calculated and observed transitions is exceptionally good. Table 25. d-d Electric Spectrum of CuC14(NH4)4Cl2. Calculateda

Observed, Ref. [72]b

∆E(eV)

Orbitals

∆E(eV)

Orbitalsc

0.94 1.24

x2-y2→ z2 xy —> z2

1.03 1.24

? ?

1.58 1.66

xz ---> z2 yz —> z2

1.58 1.62

? ?

aSCMEH-CI. bThese

results are actually for CsCuC13 which has a localized structure very similar to that of (NH4)2CuC14. There are no observed spectra reported for the ammonium compound. c The investigators in Ref. [70] assigned the orbitals in D4h symmetry, for which (according to ligand field concepts) all transitions are to the x2-y2 level. But this can be assigned with even more validity in D2h symmetry (some equatorial distortion about the 4-fold axis is indicated in the crystal structure). In this latter case, both x2-y2 and z2 belong to the totally symmetric ag irreducible representation. Thus, the observed assignments would match the calculated presented here.

Edward A. Boudreaux

H. Some lanthanide molecules One of the major difficulties with SCMEH-MO calculations on lanthanides and actinides is that the charge/configuration dependent Hamiltonian elements for the nf and (n+1)d orbitals, generate MO eigenvalues for open shells having greater stability than higher energy closed shells. In the case of samarium, for example, the Sm orbital configurations strive to be stabilized as 4f14 and/or 5d10, rather than the actual configurations 4f6 5d0 6s2, 4f6 5d0 6s0, 4f5 5d0 6s0 for Sm, Sm2+ and Sm3+ respectively. The only procedure which resolves this problem is to conduct calculations involving only the fully occupied 5s, 5p, and 6s orbitals. This allows a reasonable, converged, charge to be obtained (even with the absence of the 4f electrons) since the electron densities of 5s2 5p6 and 6s2 are outside of the 4f electrons. A second calculation is made holding the charge fixed to that of the initial calculation, but including 4f, 5d, 6s and 6p orbitals. The final assignments of electrons to all orbitals incorporated in the calculation are corrected for electron repulsion correlation. For example,\in the case of an Sm 4f6 configuration, electron repulsion energies are calculated for a 4f14 vs. 4f6 plus 4f5 5d1 (averaged) and 5d10 vs. 4f5 5d1 configurations. Finally, a third calculation is made including the electron repulsion correlation from the second calculation, followed by spin-orbit and ligand field effects and configuration interaction incorporated into all the resulting MOs. Hence, the preference of labeling this routine as SCMEH-RCCI, i.e. the SCMEH MO procedure with electron repulsion and configuration interaction included. All required atomic data are derived from QR-HF-SCF atomic calculations utilizing the programs of Klobukowski [53].

1. Sm(Cp*)2 [71], Sm(Cp*)+, Sm(Cp*)2+ [72] The bis(pentamethylcyclopentadienyl) metal compounds, M(Cp*)2, all contain the (Cp*)- anion bonded to the mewtals, M. When M is a typical dblock transition metal, the Cp*---Mâ&#x20AC;&#x201D;Cp* bonds are geometrically linear with the M bonded perpendicular to the centroid of the Cp* rings, thus maintaining the Cp* in parallel orientation.However, even large scaled ab initio calculations have been incapable of reproducing accurate M---Cp* bond distances [73]. But in the case of the lanthanide, Yb(II), the geometric configuration of the Cp* ligands are bent with respect t to the Yb(II), both in sdolid and gas phases [74]. Similarly, Sm(Cp*)2 has ma bent geometry in the solid [75], but no gas phase data are available. In conducting SCMEH-RCCI calculations on Sm(Cp*)2 in both bent (point group C2v) and linear (point group D5h) geometries, it is significant to

Theoretical foundation and applications

report that the electron repulsion correlation is very large in both cases. For Sm (II) having a net charge of +1.47, the correlation energies are 36.426 and 20.118 Ev in the m4f and 5d levels respectively. Complete data are provided in Tables 26 and 27. From Tables 26 and 27 it is noted that the filled HOMOs are essentially non bonding (in the covalent sense), since they are comprised of primarily Cp* character. The 4f6 manifold of levels for Sm(II) arte singularly occupied, forming a spectroscopic spin heptet state, which reside about 4.4eV above the filled MOs. The LUMO is of totally 4f character, while the next to lowest unoccupied MO (NLUMO) is only 0.22eV above the LUMO and has 97% Cp* character. This feature reflects a high degree of iconicity in the Sm— Cp* bonding. With the Sm to ring centroid distance, 5.27 au, and the closest Cp*—Cp* contacts 6.62 au, the total electrostatic energy is calculated to be 80.7 kcal/mole. There is no reported bond energy for this compound, but measurements on several lanthanide pentamethylcyclopentadienides yield Ln—Cp* bond energies ranging from 78 to 82 kcal/mole [76]. Additional SCMEH-RCCI results for Sm(Cp*)+ and Sm(Cp*)2+ are presented in Tables 28 and 29. In an effort to obtain a more meaningful assessment of covalency in the occupied MOs, the covalency bonding index, CBI, and a fractional covalency factor, fc, per MO, are defined as follows

CBI =

∑| C C

i , j (i ≠ j )

| Pij

(1)

where Cii, Cjj are coefficients of all AOs within an MO having a Lőwdin populations Pij. The fc are given as N/4 |Ci Cj| (CBI), where N is the MO occupancy and Ci, Cj are the coefficients. The total fractional covalency in the bonding, Fc, is the sum of fc for all occupied MOs, n, as shown in equation (2) Fc =

∑( f n

) = ∑[

c n

N | C i C j | (CBI )]n 4

(2)

A singularly occupied MO (N=1) is considered to be half-bonding. As noted, for example, in Table 28, the data reflect among the highest doubly occupied MOs (27 thru40) there is 67.7% 4f and 32.3% Cp* character in the bonding. But the six singularly occupied MOs have the AO compositions with 5/6 on Sm(1/6 5d, 4/6 4f) plus 1/6 Cp*. This is considerably different from what was found for Sm(Cp*)2.

Edward A. Boudreaux

Table 26. MO Parameters for Sm(Cp*)2ÂŞ.

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Table 27. MO Characteristics of Sm(Cp*)2ÂŞ.

Table 28. MO Parameters for Sm(Cp*)+.

Edward A. Boudreaux

Table 29. MO Parameters for Sm(Cp*)2+(a).

Interestingly, the situation in Sm(Cp*)3+ is found to be quite different from that of Sm(Cp*)+. As shown in Table 29 doubly occupied MOs are 98% localized on Sm (92% 4f, 6% 5d) and only 2% on Cp*, All singularly occupied levels are composed of Sm 81.2% 4f and 18.8% 5d orbitals. Based on equation (2) the covalency in Sm(Cp*)+ (Fc =0.360) is 2.2 times greater than that of Sm(Cp*)2+ (Fc = 0.168).

2. An (Sm4)10+ cluster The same type of calculation carried out on the Sm/Cp* systems above were extended to an Sm4 cluster contained in the complex Sm4 (N2H2)2 (N2H3)4 (NH3)2 (Cp*)4 [77]. The interesting thing about this complex are the Smâ&#x20AC;&#x201D;Sm bond distances. There are two long (4.54 angstroms) and two abnormally short (3.55 angstroms) bonds. The geometry is an elongated tetrahedron in which the Sm atoms are bridged by the nitrogen hydrido ligands, holding the Sm4 cluster intact. Each Sm is in a formal +3 oxidation state, for which there are five 4f electrons in the isolated ion. This is found to be preserved also in the Sm4 cluster complex. Details of the initial

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calculations are presented elsewhere [78]. Subsequent refinement (unpublished) to the results provided in reference [78], show that all 20 electrons residing on the four Sm atoms are unpaired in 4f orbitals. The net atomic charges are; Sm = 2.54 each; four N (in N2H2)2 = -0.77 each; eight N (in N2H3)4 = 0.41 each; four Cp* = -0.91 each. Hence the situation is such that the Sm4 cluster of ions is stabilized via ionic bonding to the bridging N ligands and the Cp* groups. In spite of the two close Sm—Sm bonds there is no covalent bonding in this complex.

3. Lanthanide carbonyls In 1973 a report was published that various lanthanide carbonyl complexes had been prepared and their infrared spectra recorded in anAr gas matrix at 10K temperature [79]. The species Nd(CO)6 was confirmed as the primary product. While carbonyls of transition metals are well known and have been the object of numerous theoretical studies of their electronic structures and bonding, those of lanthanides have not even been considered. The reason is that lanthanides are atypical and do not participate to any significant covalent bonding, as do transition metals. The ground state of Nd is 4f46s2 at the valence level. These six electrons plus the pair from each CO ligand in the hexacarbonyl molecule, gives an effective atomic number, EAN =18. This 18 electron rule, as reflected in dblock transition metal carbonyls, does indeed provide stable metal carbonyl molecules. But since Nd is atypical it is most interesting to find that its carbonyl compound even exists at all, since the bonding characteristics of lanthanide compounds are essentially ionic while carbonyls are not. Hence, an investigation of the electronic structures and bonding in Nd(CO)6 and Sm(CO)6 has been made via the SCMEH-RCCI method. It was anticipated that due to the anticipated nature of CO—Ln bonding which cannot be tonic in the normal sense, tha the Ln (Nd, Sm) atoms should acquire a small negative charge from the CO donor ligands. It was also found in prior calculations on both d-block transition metals and f-block lanthanides, that inclusion of the “n” level valence core, ns2np6, in transition metals and (n +1)s2 (n + 1)p6 in lanthanides, is desirable in accounting for the not insignificant influence that the electrons in these orbitals have on the accuracy of the calculations. Hence, the atomic basis set consisted of 5s,5p,4f,5d,6s,6p in all feasible orbital configurations with charge variation energies for each Lnq state in the range -1≤ q≤ +1. These data were derived from QR-HFSCF calculations described previously, plus reported spectroscopic data as available. During the course of the calculations bond

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distances were varied in increments of 0.06au in the range of 4.35 – 4.91au; a complete cycle of calculations made at each incremental stage, until a self consistency in Lőwdin populations and minimum energy was obtained. The optimized M—CO bond distances were 4.54 and 4.48 au for M = Nd and Sm respectively, while the C—O distances were fixed at 2.67 au. The final eigenvalues were corrected for spin-pairing, spin-orbit and ligand field effects are presented in Tables 30 and 31. Those MO levels in the -24 to -81 eV range containing the greatest contributions of 5s, 5p levels, have not been included in these tables. Note in Table 30 that the HOMO for Nd(CO)6 is a triply degenerate t2g orbital comprised of some 26 % Nd 5d and 74 % CO character. The LUMO is only 0.938eV above the HOMO and is completely 4f character. There is in addition, what might be regarded as formal bonding, a surprising degree of Nd 5s, 5p involvement. In the case of Sm(CO)6 the situation is surprisingly rather different. The HOMO is a doubly occupied a2u orbital totally 4f in character. The next MO 0.968 eV below the HOMO is a fully occupied t2g MO with some 26 % 5d and 74 % CO character. The LUMO is 0.991 eV above the HOMO and is some 92 % 4f character. Note that the net negative charge on Sm in Sm(CO)6 is about half of that on Nd in Nd(CO)6. Thus it appears that Nd is the better electron acceptor from CO and should be the more stable of the two. The calculated M—CO bond energies were 61 and 49 kcal/mole for M=Nd and Sm respectively, thus confirming the expectation that the Nd— CO bond is somewhat more stable than that of Sm—CO. But these energies are some 3 – 3.7 times lower than bond energies of a typical 5d transition metal CO bond, such as W(CO)6 [80]. For comparative purposes a DFT calculation was carried out on Nd(CO)6 [81] using the Gaussian basis Nd functions derived by Van Piggelen, Niewpoort and Van de Velde [ 82]. However, no spin-orbit effects were accounted for. Although this calculation did converge, it yielded quite unacceptable results. The main difficulty was the density matrix breaking symmetry, with all MOs being transformed into doubly degenerate and non degenerate levels. Furthermore, the converged results of this DFT calculation yielded an unrealistic, large positive charge, of 1.92 on Nd. None the less, the involvement of Nd 5p bonding was also reflected in these results. Consequently, it may be concluded that these lanthanide carbonyls do show some interesting bonding features, in contrast to d-block transition metal carbonyls, but are indeed unstable molecules, as expected.

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Table 30. MO Parameters for Nd(CO)6.

Edward A. Boudreaux

Table 31. MO Parameters for Sm(CO)6.

4. NdO vs UO The NdO molecule has been characterized as an unstable species, similar to UO, yet their electronic spectra have been studied in noble gas matrices at low temperature [83-84]. Both NdO and UO have been investigated via the QR-SCMEH-MO method [85] .The orbital bases were: Nd = 5s, 5p, 4f, 5d, 6s, 6p and U = 6s, 6p, 5f, 6d, 7s, 7p. All input data were derived from QR-HF SCF atomic calculations, as described previously. The correlation was improved by mixing the nf (n + 2)s orbitals of Nd and U in ratios of 16% and 28 respectively and mixing the O 2s and 2p in 33% and 67% respectively. Interpolated bond distances were fixed at 3.390 au for NdO and 3.470 au for UO. Results for NdO are given in the following Table 32, while those for UO are in Table 33. Ionization potentials, bond dissociation energies and electric dipole moments are presented in Table 34. Note that the agreement between the calculated values from this work with experiment (where available) are quite good for both NdO and UO.

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The bonding in NdO appears to be primarily ionic, as anticipated. More specifically, it is calculated to be 82.6 % ionic and 17.4 % covalent, However the covalency may be somewhat misleading, since the electron density plots do not show any actual sharing of density between the Nd and O inter-nuclear region, but reflects a polarization from the Nd toward the O atom [85]. The case for UO is entirely different. The bonding is calculated to be only 29.2 % ionic and 70.8 % covalent. In this instance the electron density maps do show a sharing of density between the U and O atoms [85]. Note that the ionic charges are 43% lower in UO than in NdO (see Table 34). The relativistic Dirac Fock- SCF results obtained by Malli on UO [87] do not appear to be acceptable. This calculation shows the molecule to be unbound by more than 1.6 eV above the atomic separation energies. Hence, the triumph of QR-SCMEH-MO results over those of ab initio relativistic SCF calculations. Table 32. MO Data for NdO.

Edward A. Boudreaux

Table 33. MO Data for UO.

Table 34. Some Properties of NdO and UO.

I. Some open-shell diatomic molecular metals The following review of homonuclear diatomic transition metal in Group VIB was previously published by Boudreaux and Baxter [88], where-in

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complete details are provided. These were QR-SCMEH-MO calculations with quasi-relativistic atomic functions, with spin orbit, ligand field and spin pairing effects included. The latter applies to all MOs occupied by more than one electron.The AOs included valence core ns2np6 and nd, (n+1)s, (n+1)p. The valence core involvement in these diatomic molecules is even more pronounced then what has already been discussed in other cases. This feature has been further investigated by Boudreaux and Baxter and is presented in another publication [89]. 1. Cr2 The dichromium molecule had been one of the first transition metal diatomics of theoretical interest. The most detailed ab initio calculation reported was that of Roos [90]. The experimental Cr—Cr internuclear distance is reported to be quite short at 1.679 angstroms, as compared to the metal where the bond distance is about 2.5 angstroms. This was interpreted in terms of a sextet bond involving 4s and five 3d orbitals. Roos’s calculation and those of some others, indicate that the 12 valence electrons of Cr2 are not formerly bound in the usual sense, but form a closed shell ground state via anti-ferromagnetic coupling of all 12 electrons. However, the QR-SCMEHMO results show two double (dπ, dδ) bonds, one sdσ bond and one half filled, anti-ferromagnetically coupled dδ* HOMO containing 2 electrons. Highlights of electronic structures and bonding and provided in Table 35. These results do appear to adequately describe the bonding, as evident from excellent agreement between various calculated and observed properties (see Table 36). 2. Mo2 The Mo2 molecule shows an electronic structure and bonding pattern just like that of Cr2. All 4d AOs from Mo atoms are in filled MOs, with the exception of the HOMO, a dδ* half filled level in which the 2 electrons are anti-ferromagetically coupled. This latter energy, -1.298eV, is some 5.4 time greater than that of Cr2. The key features are in Table 35 and the calculated bond energy and ionization potential are given in Table 36. 3. W2 MO features of diatomic tungsten are described in relativistic terms, since relativistic spin-orbit effects scrambles the final energies in a manner leading to a very different orbital pattern from those of Cr2 and Mo2.

Edward A. Boudreaux

Table 35. MO Highlights of M2 Molecules (M = Cr, Mo, W, Sg).

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Table 36. Bond Energies and Ionization Potentials of M2 Molecules (M = Cr, Mo, W, Sg).

The valence core has in addition to the 5s2,5p6 levels, the 4f14 level as well. It is the latter which has the primary influence in W2 and accounts for 26% off the total bonding. There are two sigma bonds, one primarily 6s and the other is the HOMO with 5dz2 character. The other 8 valence electrons are assigned four each to one 5dĎ&#x20AC; and one 5dÎ´ orbital. Note that there are no incompletely filled MOs in which anti-ferromagnetic coupling may be involved. These details are to be found in Table 35. As shown in Table 36, the calculated bond energy is in excellent agreement with the experimental, but there is no experimental ionization energy with which to compare the calculated value. 4. Sg2 The diatomic seaborgium molecule has of course never been characterized, since Sg is so highly radioactive. However, since it completes the VIB group of transition metals it is interesting to include it in this study. Here again, as in W2, the most influential part of the valence core is the 5f level. With the exception of more enhanced relativistic splitting, the bonding pattern is similar to that of W2 (see Table 35). But unlike W2 the 7s orbitals form the HOMO for Sg2; yet, there are two single and two double bonds as in W2. The calculated properties in Table 36 have no experimental data for comparison. Additional QR-SCMEH-MO calculations have been extended to two other six valence electron atoms, Nd and U, forming diatomic molecules, Nd2 and U2. However, the valence electrons in these cases are from nf, (n+2)s

Edward A. Boudreaux

orbitals, as opposed to the d orbitals of transition metals. Since the nf orbitals are deeper than the nd orbitals of transition metals, fâ&#x20AC;&#x201D;f overlap is more suppressed than that of dâ&#x20AC;&#x201D;d and the bonding in Nd2 and U2 are expected to be weaker than that of diatomic transition metals. 5. Nd2 The quasi-relativistic basis AOs are the same as reported previously for other Nd molecules, with spin-orbit and spin-pairing effects included. The inter-nuclear bond distance was optimized to R=6.206 au, which is very close to the sum of reported Nd covalent radii (6.20 au). The calculated results are in Table 37. Table 37. MO DATA for Nd2 and U2.

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The results for Nd2 presented here are very different from that reported in an MCSCF calculation [94]. In the latter case, the optimized bond distance is 9.14 au and there are 6 electrons confined to non –bonding 4f orbitals, with the remaining 6 electrons assigned as: σ2g σ1u σ1g π2u. Note that this allows only 2 sigma bonding electrons, while the other 4 electrons are in open shells. The calculated bond energy is 0.31 eV. The QRSCMEH-MO results have only 2 electrons in a half- filled phi orbital, while the other 10 electrons are in filled orbitals. There are a total of 5.5 bonds with a calculated bond energy of 0.749 eV, which compares favorably with the reported experimental value 0.87 ± 0.3 eV. The most surprising thing about the results of this study is that the population of the 4f orbital increases from 4 to 6.033, thus reflecting significant stabilization of f electrons in the molecule. The 6s orbital, on the other hand, is destabilized from 2 to 0.05. Furthermore the magnetic coupling is ferromagnetic for the 2 electrons in the open shell, having an energy of 0.635 eV, 7% of the total orbital energy. 6. U2 Interestingly, the results for U2 are essentially a duplicate of those obtained for Nd2.The internuclear bond distance R = 5.35 au, which again is very close to the sum of the U covalent radii. There are 5.5 bonds with a half-filled phi HOMO in which the 2 electrons are ferromagnetically coupled with an energy of 0.158 eV, only 1% of the orbital energy. The calculated bond energy is 2.220 eV, while the experimental value is 2.31 ± 0.22 eV. Another CASSCF ab initio calculation produced very different results [95]. In this latter case the MO scheme is: (5f1) (5f1) 7sσ2 6dπ4 6dσ1 6d1 5fδ1 5fπ1. The calculated bond energy is 1.75 eV (no spin-orbit) and 1.33 eV (with spin-orbit). Obviously this is not as good a result as that obtained by the QRSCMEH-MO method. There are several questions regarding the results of the CASSCF calculation. In the first place, overlap of the 6d—6d orbitals at a bond distance 5.353 au is 0.163. At the CASSCF bond distance, 4.59 au, the 6d overlap should be even greater. Hence, a normal 2-electron covalent bond should dominate this MO. However, the CASSCF result assigns two nonbonding electrons to the 6d level, which is not at all reasonable. It is reasonable to expect a 6d sigma bond with 2 electrons. Secondly, Based on Hund’s Rule, it is difficult to believe that doubly degererate 6dδ, 5fδ and 5fπ MOs would be occupied by only one electron each. On the contrary, these arbitals should be at least half-filled with 2 electrons, if not completely filled

Edward A. Boudreaux

with 4 electrons, in order of lowest to highest energy. In the third place, the 7s orbital of the U atom is almost twice as high in energy as the 5f. Even though the 7s/7s overlap will be greater than the 5f/5f, the energy factor plays the dominant role in the ordering of MO energies. Hence, the 7s sigma MO should be the highest MO in the 5f,6d,7s sequence and it should not be filled after the other 5f and 6d MOs. Finally, if the CASSCF results were reliable, the order of MOs should be something like: 5fσ2 5fπ4 6dσ2 7sσ2 5fδ2 or some similar scheme, but in no case should there be more open shells than closed ones.

IV. Conclusion comments and acknowledgments It has been shown the SC-MEH MO method is a reliable molecular orbital computational method, which is applicable to a wide variety of molecular systems. The superiority of this method over DFT is clearly evident when addressing open shell heavy metals and other cases as well. We extend our gratitude to Elsevier Science Publishers for granting permission to duplicate much of the content of this contribution, originally appearing in “Vibration Spectra and Structure, vol.20, 1993, pgs. 189-238. Finally, the author is forever indebted to Eric Baxter, Dr. Thomas Carsey, Dr. A. Dutta-Ahmed, Dr. Eleanor Elder and Dr. Leonce Harris for their invaluable contributions (in collaboration with the author) toward the development and applications of the SC-MEH MO method over the past forty three years. Also, the assistance afforded thru collaboration with Profs. Louis C. Cusachs and Mariuz Klobukowski cannot by any means be minimized. The assistances of the University of New Orleans, Department of Chemistry and the Computer Research Center are most gratefully appreciated for their providing facilities to accomplish this work.

V. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

a) A. Dutta-Ahmed and E. A. Boudreaux, Inorg. Chem., 12, 1590 (1973), b) ibid., p. 1597. L. E. Harris and E. A. Boudreaux, Chem. Phys. Lett., 23, 434 (1973). L. E. Harris and E. A. Boudreaux, Inorg. Chim. Acta., 2, 245 (1974). T. P. Carsey and E. A. Boudreaux, Theor. Chim. Acta., 56, 211 (1980) T. P. Carsey and E. A. Boudreaux, Chem.-Biol. Interact, 30, 189 (1980). T. P. Carsey and E. A. Boudreaux, Int. J. Quantum Chem., 18, 469 (1980). E. A. Boudreaux, Inorg. Chim. Acta.., 82, 183 (1984). E. A. Boudreaux, S. P. Doussa, M. Klobukowski, hit. J. Quantum Chem., 20, 239 (1986). E. Baralt, E. A. Boudreaux, J. N. Demas, P. G. Lenhert, C. Mj. Lukehart, A. T.

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10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42.

McPhail, J. B. Myers, Jr., L.-A. Sacksteder and W. R. True, Organometallics, 8, 2417 (1989). Ui-Rak Kiam, Sang-Hae Kim and E. A. Boudreaux, J. Korean Chem. Soc., 34, 331 (1990). E. A. Boudreaux and E. Baxter, Mol. Cryst. Liq. Cryst., 1984, 349 (1990). E. A. Boudreaux and E. Baxter, Int. J. Quantum Chem., 34, 699 (1991). L. C. Cusachs, Int. J. Quantum Chem., 1, 423 (1967). a) G. C. Allen, G. A. M. El-Sharkawy and K. D. Warren, Inorg. Chem., 10, 2538 (1971); b) G. C. Allen and T. D. Warren, in Structure and Bonding, 9, 49 (1971). F.L.M.A.H. De Laat, Doctoral Thesis, Eindhoven, The Netherlands, 1968. R. F. Fenske, K. G. Coulton, D. D. Radtke and C. C. Sweeney, Inorg.. Chem., 5, 960 (1966). J. W. Moskowitz, C. Hollister and C. J. Hornback, J. Chem. Phys., 53, 2570 (1970). K. Knox, R. G. Shulman and S. Sugano, Phys. Rev., 130, 512 (1963). P. 0. Offenhartz, J. Chem. Phys., 47, 2951 (1967). P. 0. Offenhartz, J. Am. Chem. Soc., 91, 5699 (1969). D. L. Wood, J. Ferguson, K. Knox and J. F. Dillon, J. Chem. Phys, 39, 890 (1963). C. K. Jorgensen, Absorption Spectra and Chemical Bonding in Complexes, Pergamon Press, London, 1962. W. Marshall and R. Stuart, Phys. Rev., 123, 2048 (1961). A. Abragam, Principles of Nuclear Magnetism, Clarendon Press, Oxford, 1962, p. 172. R. G. Shulman and S. Sugano, Phys. Rev., 130, 506 (1963). G. N. LaMar, Inorg. Chem., 10, 2633 (1971). D. E. Rimmer, J. Phys. A., 2, 329 (1969). A. J. H. Wachters, Doctoral Thesis, Groniryen, The Netherlands, 1971. L. Helmholz, A. V. Grizzo and R. N. Sanders, J. Chem. Phys., 135, 1349 (1961). T. P. P. Hall, W. Hayes, R. W. H. Stevenson and J. Wilkens, J. Chem. Phys., 38, 1977 (1963). R. D. Brown and P. G. Burton, Theor. Chim. Acta, 18, 309 (1970). L. Helmholz, J. Chem. Phys., 31, 172 (1959). N. Fonnesbeck, J. Hjortkjaer and J. Johansen, Int. J. Quantum Chem., 12, (Supplement 2), 95 (1977). J. P. Grima, F. Chaplin and G. Kaufman, J. Organomet. Chem., 129, 221 (1977). L. A. Hanlan, H. Huber, E. P. Kundig, B. R. McGarvey, and G. A. Ozin, J. Am. Chem. Soc., 97, 7054 (1975). A. D. Bacon and M. C. Zerner, Theoret. Chem. Acta., 53, 21 (1979). J. S. Tse, Chem. Phys. Lett., 65, 371 (1979). R. L. Sweany, Inorg. Chem., 21, 1752 (1982). J. W. Rathke and H. M. Feder, J. Am. Chem. Soc., 100, 3623 (1978). J. A. Tossell, Chem. Phys. Lett., 65, 371 (1979). a) M. C. Bohm, Z. Naturforsch., 36A,1205 (1981); b) ibid., p. 1211. S. Larsson, Physica Scripta., 21, 558 (1980).

Edward A. Boudreaux

43. M. F. Guest, B. R. Higginson, D. R. Lloyd and I. H. Hillier, J. Chem. Soc., Faraday Trans., 2, 71, 902 (1975) and footnote on p. 905. 44. D. C. Calabro and D. L. Lichtenbergen, Inorg. Chem., 19, 1732 (1980). 45. S. Cradock, E.A.V. Ebsworth and A. Robertson, J. Chem. Soc. Dalton Trans., 22 (1973). 46. J. E. Huheey, "Inorganic Chemistry Principles of Structure and Reactivity", Harper and Row, New York, 3rd Ed., 1983. Appendix E (A-32-40). 47. D. M. Cooper and S. R. Langhoff, J. Chem. Phys., 74, 1200 (1981). Also, Ref. [47]. 48. S. F. Ashcroft and C. T. Mortimer, "Thermochemistry of Transition Metal Complexes ", Academic Press, New York, 1970, Table 8-26, p. 303. 49. ibid, pp. 376-378. 50. R. L. Sweany, Dept. of Chemistry, UNO, states summary of available kinetic data on HCo(C0)4 suggest an average value of 1.55±0.19eV for Co-CO bond energy. 51. L. J. Kirshenbaum, J. M. Howell and A. R. Rossi, J. Am. Chem. Soc., 100, 3623 (1978). 52. R. D. Cowan and D. C. Griffin, J. Opt. Soc. Am., 66, 1010 (1976). 53. a) L. R. Kahn, P. J. Hay and R. D. Cowan, J. Chem. Phys., 68, 2386 (1978); b) M. Klobukowski, J. Comput-Chem., 4, 350 (1983). 54. “Platinum, Gold and Other Metal Chemotheropeutic Agents”, ACS Symposium Series, Chapter 1-10 (S. J. Lippard, ed.), Am. Chem. Soc., Washington, D. C., 1983. 55. ibid. Chapter 11. 56. P. Stein, M. K. Dickson, and D. M. Roundhill, J. Am. Chem. Soc., 105, 3489 (1983). C. M. Che, L. G. Butler, H. B. Gray, R. M. Crooks, and W. H. Woodruff, J. Am. Chem. Soc., 105, 5492 (1983). 57. W. A. Fordyce, J. G. Brummer, and G. A. Crosby, J. Am. Chem. Soc., 103, 7061 (1981). 58. S. F. Rice and H. B. Gray, J. Am. Chem. Soc., 105, 4571 (1983). 59. P. R. D. Felomena, P. J. Sadler, S. Neidle, M. R. Sanderson, A. Subbiah, and R. K. Kuroda, J. Chem. Soc. Chem. Commun., 1980, 13. 60. H. Isci and W. R. Mason, Inorg. Chem., 24, 1761 (1985). 61. M. K. Dickson, W. A. Fordyce, D. M. Appel, K. Alexander, P. Stein and D. M. Roundhill, Inorg. Chem.., 21, 3857 (1982). 62. Y. Shimizu, Y. Tanaka and T. Azumi, J. Phys. Chem., 88, 2423 (1984). 63. M. Shah, K. Thangaraj, M-L, Soog, L. T. Wolford, and J. Boyer, Heteroatom. Chem., 1, 389 (1990). 64. C. L. Picou, E. D. Stevens, M. Shah and J. H. Boyer, Acta. Cryst., Soc. C., 46, 1148 (1990). 65. E. A. Boudreaux, K. Chin and J. Seminario - Publication in preparation. 66. Unpublished X-ray structure by E. D. Stevens and Co-workers, shows the isopropyl hydrogen at right angles to the plane of the ring. Boyer and associates did not observe any fluorescence for this compound. 67. R. D. Willett, J. Chem. Phys., 41, 2243 (1964).

Theoretical foundation and applications

68. P. C. de Mello, M. Hehenberger, S. Larsson and M. Zerner, J. Am. Chem. Soc., 102, 1278 (1980). 69. Calculation performed through the courtesy of Jorge Seminario, University of New Orleans, Department of Chemistry. 70. R. Laiko, M. Natarajan, and M. Kaira, Phys. Status Solidi A, 15, 311 (1973). 71. Boudreaux,E.A. and Baxter,E. Int. J. Quantum Chem., 28, 565 (1994). 72. Boudreaux,E.A. and Baxter, E.Int. J. Quantum Chem., 29, 605 (1995). 73. Luthi, H.P., Ammeter, J. H., Almolof, J. and Faegi, K.J. Chem. Phys., 77, 2002 (1982) 74. Anderson, R.A., Barcella, J.M., Burns, C. J., Haalaud, A. and Volden, H.V. J. Organometal. Chem., C49, 312 (1986); Acta. Chem.. Scand., A41, 24 (19087); Chem. Soc. Chem. Commun., 768 (1987). 75. Evans, J.W., Hughes, L.A. and Hanera, T.P. Organometallics, 5, 1285 (1986). 76. Harlley,F.R. and Patin,S., “ The Chemistry of the Metal-carbon Bond”, John Wiley and Sons, London,1982, Chap.2. 77. Wang, K-G, Stevens, E.D. and Nolan, S. P. Organometallics, 11, 1011 (1902). 78. Boudreaux,E.A. and Baxter, E. Int. J. Quantum Chem., 64, 297 (1907). 79. Slater, J.L., DeVore, T.C. ande Calder, V. Inorg. Chem., 12, 1918 (1973). 80. Connor, J. A., Topics Current Chem., 1971, 71. 81. Boudreaux, E.A. and SEminario, J.M. J. Mol. Struct. (Theochem), 425, 25 (1998). 82. Van Piggelen, H.V. Nieupoort and Van der Velde, G.A. J. Chem. Phys.’ 72, 3727 (1980). 83. Kaledin, L. A., PhD Thesis, Emory University (1985).Private communication. 84. Kaledin, L.A., McCord, J.E. and Heaven, M.C. J. Mol. Spec.,164, 27 (1994). Links. 85. Boudreaux, E.A. and Baxter, E. C. Int. J. Quantum Chem.,90, 629 (2002) 86. Goldstein, H.W., Walsh, D.W. and White, D.J. Phys. Chem.,65, 1400 (1960); Walsh, P.N. DEver,D.E. ibid., 1410. 87. Malli, G.L.”The Challenge of d and f Electrons, Theory and Computation”, Salahub, D.R. and Zerner, M.C. Eds.ACS Symposium Series, Am. Chem. Soc., Washington, D.C., 1989, Chap. 21, 301. 88. DeMaria G., Burns, R.P. Drowart, J. and Ingram, M.G.J.Chem. Phys.,32, 1373 (1960). 89. Boudreaux, E.A.and Baxter,E. Int. J. Quantum Chem.,100,1170 (2004). 90. Roos, B.O. Collect. Czech. Chem. Commun.,68,265 (2003). 91. Simard, B., Leberault-Dorget, M.A., Marinissen, A and ter Meulen, J.J.J. Chem. Phys., 108, 9668 (1998). 92. Morse, M.D. Chem. Rev., 86, 1049 (1986). 93. Brewer, L. and Winn, J.S. Faraday Symp. R. Soc. Chem., 14, 126 (1980). 94. Dolg, M., Stoll, H. and Preuss, HG. J. Mol. Struct. (Theochem), 0V227, 239 (1992). 95. Gaglardi, L. and Roos, B.O. Nature, 433, 848, 2005.

Transworld Research Network 37/661 (2), Fort P.O. Trivandrum-695 023 Kerala, India

Approximate Molecular Orbital Methods, 2010: 63-84 ISBN: 978-81-7895-466-0 Editor: Edward A. Boudreaux

2. Molecules in different environments: Solvatochromic effects using Monte Carlo simulation and semi-empirical quantum mechanical calculations 1

Kaline Coutinho1, Tertius L. Fonseca2 and Sylvio Canuto1

Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo, SP, Brazil Instituto de Física, Universidade Federal de Goiás, CP 131, 74001-970, Goiânia, GO, Brazil

Abstract. The sequential QM/MM methodology is used to

describe the solvent effects on the electronic absorption spectra of organic molecules in solution. The structure of the liquid is generated by Monte Carlo computer simulation. Configurations composed by the solute and several solvent molecules are selected for a posteriori quantum mechanical calculations of the spectra. Situations are considered where a large number of solvent molecules are necessary to describe the solvation problem. The examples considered here involve supermolecular systems composed of ca. 1500-2000 valence electrons, justifying the need for a semi-empirical approach. The electronic spectrum is then calculated using the INDO/CIS method. The solvatochromic shifts of pyrimidine in water and of beta-carotene in acetone and isopentane are considered. These exemplify the situations of a polar molecule in a polar Correspondence/Reprint request: Dr. Sylvio Canuto, Instituto de Física, Universidade de São Paulo, CP 66318 05315-970, São Paulo, SP, Brazil. E-mail: canuto@if.usp.br

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environment and of a non-polar molecule in both polar and non-polar environments. An additional example is considered where the absorption spectrum of acetone is analysed in a low-dense condition of a supercritical water environment. Good agreements with experimental shifts are obtained in all cases. The relative importance of the inner and outer solvation shells is analyzed. The case of beta-carotene is a persistent and difficult problem because the spectrum involves a π-π*excitation between two states of zero dipole moment. For the case of acetone in supercritical water analysis is made of the decrease in the solute-solvent hydrogen bonds and their role on the calculated solvatochromic shift. This solvatochromism is considered both in relation to the gas phase (blue shift) and the normal liquid water spectra (red shift). The success of the present approach emphasizes the importance of the combined use of quantum mechanics and statistical mechanics and the usefulness of the semi-empirical method employed.

1. Introduction Semi-empirical methods have been an important ally in the theoretical studies of ultra-violet-visible (UV-Vis) spectra [1]. Already in the early days of quantum chemistry, the very simple Hückel model [2] gave the first qualitative explanation of the complex UV-Vis absorption spectrum of the benzene molecule. Hückel model gives the correct picture: the nature of the π-π* transitions, the degeneracy of the molecular orbitals, the degeneracy of the intense and allowed 1E1u band and the origin of the three absorption transitions that are the characteristic signature of the benzene molecule. Since those days ab initio quantum chemistry has seen an unprecedented development [3]. But because of the enormous computational difficulties of ab initio methods to handle large molecular systems the semi-empirical techniques have been very important in elucidating many aspects and, in particular, absorption spectra. In spite of the extraordinary computer revolution semi-empirical methods are of great value and will certainly continue to be as the limit of interest is systematically moving forward. With the continuous developments of both computer hardware and software large molecular systems can be considered but then the interest is also slowly shifting towards larger systems, including bio-molecules, and semi-empirical methods thus remain of interest. The quantum chemistry horizon indicates that the interest in bio-molecular systems will be considerably increased in the coming years. But in addition to the increasing size of the molecular systems of interest one is also interested in situations where a molecule is not isolated from the environment. Including additional molecules similarly increases the complexity of the system and the necessity for a simple

Molecules in different environments

computational approach. This chapter deals with the application of semiempirical methods, not to analyze a single large molecule but instead to analyze the UV-Vis absorption spectra of organic molecules in an explicit solvent environment. We are particularly focusing in the situations were the solvent effects on the solute molecule require consideration of the explicit solvent molecules. This is a situation where, even if the reference molecule is of a small size, amenable to ab initio procedures, the necessity of explicitly including the solvent molecules imposes severe limitations. First-principle calculations cannot be performed, at present, for a system surrounded by ca. 100 solvent molecules. Although there are indeed situations where the solvent effects are local, such as chemical shift in NMR shielding, for instance, there are also circumstances where solvent molecules, located far from the solute, can still affect the solute properties. This is the situation we report here. The solvatochromic shift of a polar molecule is one of the possible examples. Polar molecules in polar environment lead to an electronic polarization that extends over a large distance. But even the properties of non-polar molecules are influenced by explicit solvent molecules. This seems to be the case of the red shift of the absorption spectra of beta-carotene in different solvents. But using explicitly one beta-carotene molecule surrounded by the first solvation shell of acetone molecules leads to a problem involving more than 2000 valence electrons. This is a situation where semi-empirical methods can be of great value. The theoretical procedures to study solvent effects may be classified in two major categories. The first one is the so-called continuum dielectric methods. This is based in the ideas of Kirkwood [4] and Onsager [5] that has been developed into the self-consistent reaction field (SCRF) [6-10]. Further developments have been obtained leading to the conductor-like screening model (COSMO)[11] and the polarizable continuum method (PCM) [12] and some variants [13]. The second major category is composed by the combination of quantum mechanics and statistical mechanics leading to the QM/MM methods [14-17]. The use of molecular mechanics (MM) combined with quantum mechanics (QM) is an increasing and powerful technique to deal with the effects of the environment (generally treated by MM) into a reference molecule (treated by QM). However the partition between the QM and MM parts may not be so clear and may depend on the property of interest. This aspect led to the idea of carrying the QM/MM not at the same time but in two separate steps. In this sequential QM/MM (S-QM/MM) methodology [18-20] one first performs the molecular simulation to generate the solute-solvent configurations. Statistically uncorrelated configurations are then sampled for subsequent QM calculations. This is an efficient procedure that properly used ensures statistically converged results with a relatively

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small number of QM calculations [18-26]. This procedure gives the additional advantage of flexibility in the size of the solute-solvent system to be submitted to QM calculations and also on the QM model to be used. This last is important because the determination of different molecular properties often requires different quantum chemical methods (the model and the basis set). The disadvantage is that uncoupling the MM and QM parts requires additional consideration of the solute polarization by the solvent. This has also been considered in the S-QM/MM methodology [21]. Sampling supermolecular structures signifies several configurations composed by the solute and the surrounding solvent molecules. Thus not only the system may be considerably large, but also the QM calculations have to be performed several times to give the ensemble average that characterizes the liquid properties. In some situations and for some molecular properties it is still possible to perform ab initio QM calculations [22-24] but in this chapter we reserve the examples where the use of semi-empirical methods is, at present, mandatory. In the following section we briefly describe how the solvent environment around the solute is generated in a specific liquid and in a specific thermodynamic condition. Next we also briefly discuss how the statistically uncorrelated configurations are sampled. And finally we discuss the solvatochromic shift in the UV-Vis absorption spectra of three molecules in solution. The first example is the case of pyrimidine in water [25], a polar molecule in a very polarizable solvent environment. We analyze the relative importance of the inner and outer solvent shell structures. As conventional QM/MM calculations often use solvent electrostatic contribution alone we also analyze the role of the electrostatic embedding around explicit solvent molecules. Next, we discuss the case of beta-carotene [26] in acetone and isopentane. A non-polar molecule immersed in polar and non-polar environments, respectively. Finally, we consider the case of acetone in supercritical water. This exemplifies a situation where the molecular absorption spectrum is modified by a low-dense supercritical fluid and thus can be a useful probe for the interesting physico-chemical properties of this fluid environment.

2. Method The structure of the liquid is obtained by Monte Carlo (MC) Metropolis simulation [27]. Periodic boundary conditions, with the minimum image method in a cubic box, are used. The simulations of the first two examples considered here are performed in the NVT ensemble, with a solute surrounded by N solvent molecules at room temperature (298 K). The system

Molecules in different environments

is extremely diluted and the density is that of the solvent. For pyrimidine we use 900 water molecules. For beta-carotene we use 900 solvent molecules (acetone and isopentane). The corresponding densities of the simulated systems are 0.9966 g/cm3 (pyrimidine in water) and 0.7682 g/cm3 and 0.6001 g/cm3 (beta-carotene in acetone and isopentane, respectively). The intermolecular interactions are described by the standard Lennard-Jones plus Coulomb potential with 3 parameters for each site i (ﾎｵi, ﾏナ and qi). For the water molecules we use the SPC potential [28]. For pyrimidine, betacarotene, acetone and isopentane we use the OPLS [29]. Further details of the potential and geometries are described in previous publications [25,26]. After thermalization, the MC simulation is made with typically 108 MC steps. A new configuration is generated after N MC steps, i.e., after all solvent molecules are attempted to translate or rotate around a randomly chosen axis. As successive configurations do not give significant new statistical information we calculate the correlation interval using the auto-correlation function of the energy to sample statistically relevant configurations [1820,30]. Configurations having less than 15% of statistical correlation are selected from the MC simulations for the subsequent QM calculations. An important point in this issue is that statistical convergence is obtained in all cases reported here. All the MC simulations were performed with the program DICE [31]. To select the size of the solute-solvent structures, the pair-wise distribution function [27] is used. For extended systems like beta-carotene the usual radial distribution is not appropriate and we use a minimum-distance distribution [26,32]. For each case considered we explicitly state the size and the number of solvent molecules used. The QM calculations of the absorption spectra are made using the ZINDO program [33] with the INDO/CIS parametrization suggested by Ridley and Zerner [34]. We first calculate the spectrum of the isolated molecule for the reference. Then the solvatochromic shift is obtained by calculating the spectrum of the solvated molecule including the explicit solvent molecules. These calculations lead to the average for L statistically uncorrelated configuration and the shift is thus obtained as the difference of this average and the result for the isolated situation. The wave function is anti-symmetric with respect to the entire solute-solvent system. This allows the delocalization of the wave function into the solvent region and gives some contribution to the dispersion interaction [35]. Dispersion interaction contributes to a red shift [36] and this is particularly important for non-polar solutes such as beta-carotene. Finally, we should also note that calculations with a varying number of molecules impose the use of size-extensive methods, as it is the case of singly excited configuration interaction (CIS) methods. As most low-lying excited states are

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derived from single electron promotion the use of INDO/CIS is a natural choice for a semi-empirical method.

3. Results and discussions 3.1. Pyrimidine in water We first discuss the case of pyrimidine in water. This is an interesting case of a polar molecule in a polar environment that has attracted some theoretical interest [25, 37-42]. In pyrimidine there are two proton-acceptor sites for hydrogen bonds and their contribution to the total solvatochromic shift is still controversial [41]. The experimental n→ π* transition of pyrimidine in water has been well studied before [43-45]. The experimental transition has been reported at 36900 cm-1 in water. Baba et al. [44] reported n→ π* transition in isooctane [44,45] as 34250 cm-1 . This would correspond to an isooctane-water blue shift of 2650 cm−1. Because of the small polarity of isooctane the shift from the gas phase should be only slightly larger, close to 2700 cm-1. Indeed additional analysis [37] of several experimental results suggests a blue shift of 2700 ± 300 cm−1 for the n→π* transition of pyrimidine in water compared to gas phase. This is now our reference value for the experimental solvatochromic shift. On the theoretical side, some previous efforts have been made. Zeng et al. [37] performed a systematic study using different intermolecular pair potentials. They obtained a blue shift of 2450 cm-1, in good agreement with experiment. But they also concluded that hydrogen bonding accounts for half of the observed blue shift. This is an interesting aspect and suggests that the use of explicit solvent molecules is essential for a proper treatment. They have also noted that the pyrimidine-water hydrogen bonds may have long-range influences on the solvent shift. Karelson and Zerner [38], employing INDO/CIS calculations in the dielectric continuum approach, concluded that the blue shift could only be predicted with the inclusion of two explicit water molecules making hydrogen bonds to the two nitrogen atoms of pyrimidine. After this they estimate a blue shift of 2600 cm-1. Using DFT calculations Kongsted and Mennucci [42] also used a dielectric continuum around two explicit water molecules, hydrogen-bonded to pyrimidine, to find only a small solvatochromic shift of 1600 cm-1. They suggested that both specific and bulk effects are important. Gao and Byun [39] using hybrid QM/MM Monte Carlo simulations reported a value of 2275 ± 110 cm−1 for the n→π* blue shift of pyrimidine in water. The contribution of the hydrogen bond shell, in this case is found to be dominant. Recently, Liu et al [40] considered density-functional

Molecules in different environments

theory within the self-consistent reaction field and obtained the shift of 2500 cm-1. To understand the role of the specific interaction and the influence of the outer solvent molecules we extend the theoretical analysis performing INDO/CIS calculations including the solvent as explicit molecules. Our results will discuss the influence of the different solvation shells in the solvatochromic blue shift. First, from the statistical distribution of MC configurations we find an average number of 1.3 hydrogen bonds. This is in agreement with ref [42] that reports 1.2 hydrogen bonds. A typical configuration is shown for illustration in Figure 1a. The solvatochromic shift obtained from the structures with hydrogen bonds only,

(a)

(c)

(b)

(d)

Figure 1. Illustration of the (a) hydrogen bond and the hydration shells of pyrimidine in water. The (b) first, (c) second and (d) third shells are composed of 21, 71 and 213 water molecules, respectively. The solute-solvent center of mass distances are 5.5, 8.0 and 11.6 Ă&#x2026;.

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discarding all the other water molecules, is sizable (see below). Analyzing the hydrogen-bonded complexes of pyrimidine and water Cai and Reimers [41] suggest that the contribution of the inner shell is equivalent to that of the outer shells. This would attribute a large importance to the hydrogen bond shell. The previous theoretical studies all indicate that solvent molecules beyond the hydrogen bond shell are very important. Before discussing further our results including the inner and outer shells of explicit solvent molecules it is interesting to comment that the two nitrogen atoms giving two separate n→ π* transitions are not distinguished in solution. Previous studies [38,39] have noted that including the specific water molecules that make hydrogen bonds with pyrimidine joins the two n→ π* transitions allowing intensity borrowing and leading to band broadening. Hence the results reported here are the average between the two calculated n→ π* transitions. In this circumstance the contribution of the hydrogen bond shell is 1450 cm-1. Figure 1 illustrates the solvation shells of pyrimidine in explicit water molecules. Table 1 reports the calculated results for the blue shift. It is clear from this table that including only the first solvation shell is not enough to describe the solvatochromic shift. In fact including all solvent molecules up to a distance of 5.5 Å leads to a shift of only 1600 cm-1 that is a small value compared to the experimental shift of 2700 ± 300 cm-1. It is only after including 213 explicit water molecules, corresponding to all water molecules within a distance of 11.6 Å, that the shift of 2000 cm-1 is obtained. This corresponds to a 1734 valence-electron problem. Extending the results to the bulk limit gives a solvatochromic shift of 2400 cm-1, now in good agreement with the experimental result and in line with the previous theoretical estimates. It is clear that the water molecules located in the outer solvation shells can still influence the solute spectrum. The hydrogen bonds give an additional increase of the local dipole of the chromophore leading to a longrange polarization. The results are compatible with this picture and hence we conclude that the polarization effects of polar hydrogen-bond acceptor solutes in protic solvents extend to a long distance from the solute. This is further corroborated by noting that the calculated solvatochromic shift of pyrimidine in non-polar carbon tetrachloride, is converged with the first solvation shell only [25]. This conclusion is based on large QM calculations that required the explicit consideration of the CCl4 solvent molecules up to a distance of 13.3 Å and very large QM calculations involving nearly 2000 valence electrons [25]. Clearly these calculations with explicit solvent molecules could not be made outside the scope of semi-empirical methods. The role of the hydrogen bond and the inner shells to the total solvatochromic shift of pyrimidine in water is of interest [41]. The inner and

Molecules in different environments

Table 1. Calculated blue shift of the n→ π* transition of pyrimidine in water. Shift is an average of the two n→ π* transitions. HB is the hydrogen-bond shell, NS is the number of explicit solvent water molecules in the solvation shell. M is the total number of valence electrons included in the quantum mechanical calculations. L is the number of MC configurations used for ensemble average. R is the radius of the solvation shell obtained from the radial distribution function. All calculations used the INDO/CIS method. Calculated uncertainty is the statistical error. Conversion: 1eV = 8067 cm-1.

the outer shells are important but the corresponding relative importance differs in different procedures. This has been pointed by Cai and Reimers [41] that noted that Gao and Byan [39] predicted larger contribution from the inner shell and a red shift for the dielectric contribution. This is the opposite to what we have obtained, up to this point. We obtained that increasing the number of solvent molecules increases systematically the blue shift of the n→ π* transition of pyrimidine in water. As discussed [41], other models

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have obtained otherwise [39] with the outer solvent molecules decreasing the large result obtained for the inner shells. This has led to the belief that the solute-solvent hydrogen bond alone is sufficient, or gives the most significant contribution, to the total shift. To clarify this aspect we extend the calculation now to embed the explicit solvent molecules in the electrostatic field of the remaining water molecules. The role of the hydrogen bonds and the explicit inclusion of solvent molecules will be discussed. We have then selected from the configurations generated by the MC simulations the 500 water molecules that are nearest to pyrimidine. Initially these will be treated as simple point charges and gradually will be substituted by explicit water molecules. The results are also shown in Table 1. Using only point charges for the solvent molecules (electrostatic contribution only) leads to the large value of 2970 cm-1 for the solvatochromic shift. The electrostatic contribution alone, clearly overestimates the total shift. However, gradually increasing the number of explicit molecules then decreases the calculated shift. This should be compared with the results of ref [39]. Using explicitly only the solute and the hydrogen-bonded water molecules embedded in the electrostatic field of the remaining water molecules decreases this value to 2630 cm-1, in very good agreement with the experimental shift. But this is an artifact as it can be noted by further including the outer water molecules. Explicitly using all 21 water molecules of the first solvation shell embedded in the electrostatic field of the remaining (479 treated as point charges) gives a value of 2470 cm-1, which is also a good result. Proceeding further to the largest case of 213 explicit water molecules in the electrostatic field of the remaining 287 molecules treated as simple point charge gives the value of 2100 cm-1. These results explain the red shift of the dielectric contribution that has been noted before [39, 41]. Using only the electrostatic field of the solvent into the solute molecule overestimates the solvatochromic shift and inclusion of explicit molecules is necessary to obtain a proper description. In this case increasing the number of explicit molecules decreases the shift. Including a relatively large number of explicit solvent molecules in the electrostatic field of the outer shell is potentially a very good model. But using only the solute in the electrostatic field of the solvent seems to overestimate the solvatochromic shift. Including the solute polarization by the solvent may improve the results as seen in recent application [21,32].

3.2. Beta-carotene in acetone and isopentane Carotenoids are very important in photosynthesis. The all-trans betacarotene (Figure 2) is involved in the important mechanism of energy transfer with chlorophyll. Beta-carotene absorbs visible light in the region of 450 nm

Molecules in different environments

Figure 2. The structure of all-trans beta-carotene.

giving its characteristic colour. As beta-carotene is not soluble in water and also has very low volatility the absorption spectrum has not been recorded in either water or in gas phase. The study of the solvatochromic shifts of betacarotene has then to rely on experimental results made on different solvents [46] or in ionic liquids [47]. The solvent effects on the visible spectrum of beta-carotene are a real challenge for theoretical methodologies for at least two aspects. First, the visible spectrum characterized by a strong Ď&#x20AC;-Ď&#x20AC;* absorption transition in the region of 450 nm suffers only small shifts in different solvents [46,48]. The shift from acetone to isopentane, for instance, is only 310 cm-1. The small magnitude of the shifts can be understood: the dipole moment is zero both in the ground and in the first excited state. The dominant dipolar interaction is then zero and the shift is dominated by dispersion interaction. As the dipole polarizability of the excited state is expected to be larger than in the ground state the dispersion will contribute to a better solvation of the excited state. This decreases the energy difference between the two states. This differential interaction is small for different solvents. Second, another difficulty is that beta-carotene is a relatively large molecule (ca. 30 Ă&#x2026; long), composed of 216 valence electrons, with an elongated shape imposing the use of a non-spherical solvent shell. We have developed a minimum-distance distribution function [26,32] that follows the molecular shape and can be used for any molecule, no matter how elongated or distorted. The spectrum of beta-carotene has been analyzed by Applequist [49] using a cavity model, where the chromophore has been treated as classical point dipole oscillators. Myers and Birge [48] studied the change in oscillator strength of the absorption of beta-carotene in different solvents and found that the results depend on the cavity geometry. Zerner made an estimate of the shift of beta-carotene in cyclohexane [50] using SCRF. Abe and coworkers [46] analyzed solvent effects in 51 different solvents and made an empirical analysis in terms of reaction field models. Here, we use the results obtained with the S-MC/QM methodology, in a minimum-distance solvation

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shell, to discuss the solvatochromic shift of beta-carotene in two different solvents; namely, isopentane and acetone. These two solvents are selected on the basis of their nature. Isopentane is non-polar and has a very low normalized polarity (0.006). Acetone is a polar molecule having a relatively large polarity (0.355). Using configurations obtained from the MC simulations, INDO/CIS calculations are performed on several super-molecular structures composed of one beta-carotene and NS surrounding molecules of solvent. The results are summarized in Table 2 that also gives some data pertaining to the solvents and the number NS of explicit molecules used. Figure 3 illustrates a typical configuration, extracted from the simulation, of one beta-carotene surrounded by the first shell of solvent acetone molecules. As discussed above the gas phase Table 2. Calculated absorption transitions (in cm-1) of beta-carotene in vacuum and in two different solvents. All calculations used the INDO/CIS method. NS is the total number of explicit solvent molecules and M is the total number of valence electrons included in the quantum mechanical calculations. Solvent

Dielectric constant

Normalized polarity

Transition

Experiment [46]

Vacuum

216

22230

Isopentane

1.828

0.006

2104

22180

22360

Acetone

21.36

0.355

2064

22070

22050

Figure 3. Illustration of one configuration obtained from the MC simulation. The system is composed of one beta-carotene molecules surrounded by nearest-neighbors acetone solvent molecules.

Molecules in different environments

absorption transition is not known experimentally. The value calculated here for the gas phase π-π* transition is 22230 cm-1. In solvents of any polarity this transition suffers a red shift. The magnitude of the shift, of course, depends on the solvent. Using NS = 59 isopentanes solvent molecules the average transition of beta-carotene changes to 22180 cm-1, corresponding to a red shift of 50 cm-1. In acetone the transition is obtained at 22070 cm-1, in good agreement with the experimental value of 22050 cm-1. These transition energies are obtained using 40 INDO/CIS calculations on statistically uncorrelated configurations. In the case of acetone each calculation is made on beta-carotene surrounded by 77 acetone molecules, including the explicit consideration of 2064 valence electrons, in a wave function that is anti-symmetric with respect to the permutation of any two electrons. The wave function delocalization over the solvent region, followed by the CIS calculations, contributes to the differential dispersion interaction [35] and is the main responsible for the red shift. Table 2 shows that the calculated transition energy values are in good agreement with experiment and that the solvatochromic shifts have the correct trend. In both cases the correct sign (red shift) has been obtained. But the relative magnitude is more difficulty. The red shift of the π-π* absorption transition of beta-carotene from isopentane to acetone is calculated as –110 cm-1 compared to the corresponding experimental value of – 310 cm-1.

3.3. Acetone in supercritical water It is well known that the coexistence line of the liquid and solid phases finishes at the so-called critical point. This is the point where the system becomes a supercritical fluid and exhibits physico-chemical properties that are markedly different from normal liquid systems [51,52]. Water becomes an exciting supercritical fluid at temperatures and pressures beyond the critical point located at Pc = 220 atm and Tc = 647 K. In this regime the dielectric constant is considerably decreased and water becomes an excellent solvent for many organic compounds. The density is also very much modified and under small variations of temperature and pressure it suffers intense changes. The situation is illustrated in Figure 4 that shows the density of water as a function of temperature and pressure and the location of the critical point. Understanding the properties of supercritical water (SCW) is of particular interest as water is the most important molecular system in nature. A direct approach can be made to understand the structural aspects of water by using X-ray and neutral diffraction experiments [53-57]. One aspect that emerges from experimental studies is the reduced number of hydrogen bonds as the density of water is decreased [53-58]. However the analysis of the electronic structure of supercritical water is conveniently made using a probe

Kaline Coutinho et al.

Figure 4. The density of water as a function of temperature and pressure. The location of the critical point is shown.

molecule and analyzing the change in the absorption spectrum compared to another thermodynamic situation, that is used as a reference. This reference could either be the condition of isolated molecule (gas phase situation) or the condition of normal liquid (P = 1 atm, T = 298 K). An important condition in this issue is, of course, that the probe molecule should be stable in SCW under different conditions of temperature and pressure. This is the case of acetone, where the n−π* transition in water has been studied in different supercritical conditions [59-61]. Bennett and Johnston [59] have made systematic experimental studies of different SCW conditions. From this work it is possible to characterize that for P = 340.2 atm and T = 673 K the n−π* transition of acetone suffers a blue shift of 500-700 cm-1, compared to the gas phase. In addition, there is indirect evidence [23,60,61] that the number of hydrogen bonds between acetone and water is reduced and that these are responsible for half of the total blue shift. Different thermodynamic conditions can be studied theoretically and in fact a recent theoretical analysis [23,61] of this blue shift can be found. In this section we now analyze the reduction in the number of solute-solvent hydrogen bonds, their participation in the spectral shift and, finally, the role of the inner and outer solvation shells for describing the total spectral blue shift. To obtain the solute-solvent configurations we use the MC simulation but now, as the pressure as well as the temperature are the important

Molecules in different environments

thermodynamic parameters, we have used the NPT ensemble. To compare with the experimental condition described above the MC simulations are made using P = 340.2 atm and T = 673 K. The system consists of one acetone molecule surrounded by 700 water molecules. For water we now use the SPC/E potential [62], as it correctly describes the critical point of water [63,64]. The calculated density is 0.46 g/mL. Comparing with the density of water at the critical point (0.32 g/mL) shows that this corresponds to the nearcritical regime (0.5 ≤ ρ/ρc ≤ 1.5). A density of 0.46 g/mL is expected to exhibit sizable changes in the number of hydrogen bonds. This is analyzed next. The identification of solute-solvent hydrogen bonds are normally made by considering the radial distribution function that characterizes the pair-wise atomic distances. Figure 5 show the calculated radial distribution function between the oxygen atom of acetone and the hydrogen atom of water. A clear structure is seen centered at 1.85 Å, starting at 1.50 Å and ending at 2.55 Å. This corresponds to the hydrogen-bond configurations between the acetone and water molecules. Although it is normally correct that the hydrogen bonds are found in this geometric region it cannot be assured that all water molecules located in this region are indeed hydrogen bonded to the solute. An additional consideration is made regarding the interaction energy between the solute and the solvent. Figure 6 shows the pair-wise energy distribution and

Figure 5. The pair-wise radial distribution between the O atom of acetone and the H atom of water for the supercritical condition.

Kaline Coutinho et al.

Figure 6. The pair-wise interaction energy between acetone and water for the supercritical condition.

the bump corresponding to the hydrogen bonding energies. Combining the results given in Figures 5 and 6 we obtain an average number of 0.7 hydrogen bonds between acetone and water. This number is indeed reduced compared to the case of water in normal thermodynamic condition that gives a value of 1.6 using the same type of analysis [23]. As it has been thoroughly discussed before one of the important aspects of SCW is the reduced number of hydrogen bonds. We now discuss the statistics of hydrogen bonds formed between acetone and SCW. The calculation indicates that 42.0% of the configurations make no hydrogen bonds. But 49.2% make one configuration. Proceeding, 8.5% of the configurations make 2 hydrogen bonds and a very small number (0.3%) make even 3 hydrogen bonds. The statistics thus implies that the most probable number of hydrogen bonds is simply one. But the average number is 0.7. Figure 7 shows in a single picture the superposition of all configurations that exhibit acetone-water hydrogen bonds. This superposition shows the configuration space that is spanned by the neighboring water molecules that are involved in hydrogen bonds (HB). We now analyze the contribution of the different hydration shells to the total calculated blue shift of the nâ&#x2C6;&#x2019;Ď&#x20AC;* transition of acetone in supercritical water. First, we analyze the role of the HB shell. It has been indirectly estimated that this contributes to half of the total solvatochromic blue shift. Table 3 shows the results. As it can be seen using the configurations that make HB we obtain a solvatochromic shift of 330 cm-1, compared to the

Molecules in different environments

Figure 7. Superposition of the configurations showing hydrogen bonds between acetone and supercritical water. The remaining water molecules are removed for clarity and for explicitly showing the configuration space spanned by hydrogenbonded water molecules. Table 3. Calculated solvatochromic shift (cm-1) of the n→ π* transition of acetone in supercritical water (P = 340.2 atm, T = 673 K). Results show the calculated blue shift compared to the gas phase and the red shift compared to normal water. All calculations used the INDO/CIS method. HB is the hydrogen-bond shell, NS is the total number of explicit solvent molecules and M is the total number of valence electrons included in the quantum mechanical calculations. Solvatochromic shifts were obtained as averages over 100 uncorrelated configurations. Solvation shell

Blues shift (Gas phase)

Red shift (Normal water)

(0,1,2,3)a

330

450

First

264

630

570

Second

100

824

660

700

Third

170

1384

670

730

600 ± 100

800 ± 200

Experiment [57] a)

Average number of hydrogen bonds is 0.7. See text.

Kaline Coutinho et al.

experimental result of 600 ± 100 cm-1. This is indeed in excellent agreement with the expectation that the HB shell contributes to half of the shift. The next shell gives an additional contribution and the total shift obtained using 30 explicit water molecules is 630 cm-1, showing that the first hydration shell is a good approximate model for obtaining the total shift. This is likely to be a consequence of the reduced density of water in this SC condition. Using next the second and third shells improves only slightly the result and gives our best estimate of 670 cm-1, in excellent agreement with the experimental result. The largest calculation, using explicitly 170 solvent water molecules involves a total of 1384 valence electrons. This situation is illustrated in Fig. 8, where all water molecules within the center of mass distance of 11.0 Å are explicitly included in the INDO/CIS calculations.

Figure 8. Illustration of acetone immersed in supercritical water. This corresponds to all 170 water molecules located within 11.0 Å.

Molecules in different environments

In agreement with experiment the solvatochromic shift of the of the n−π* transition of acetone in supercritical water is calculated to suffer a blue shift of 670 cm-1 compared to isolated acetone, and a red shift of 730 cm-1 compared to normal water. As inferred experimentally, the blue shift from gas phase to water is reduced for the SCW condition from 1500 ± 200 cm-1 (normal water) to 600 ± 100 cm-1. This reduction is rationalized to be one consequence of the reduced number of hydrogen bonds. In fact, it can be noted that 50% of the total shift derives from the configurations that exhibit hydrogen bonds between the solute and the solvent. However, as derived from the results of our calculations, shown in Table 3, this is not peculiar to the SCW condition and in fact also happens for normal water.

4. Summary and conclusions A combined use of Monte Carlo simulations and quantum mechanics calculations are made to analyze the absorption spectra of organic molecules in different solvent environments. The MC simulation generates the structure of the liquid to be used in QM calculations of the spectrum. We focus on the solvatochromic shifts associated to different solvents. Using super-molecular structures composed of the solute and several solvent molecules we have analyzed the role of the explicit consideration of the solvent molecules. This leads to fairly large systems imposing the consideration of semi-empirical approaches. Typically the systems considered here involve ca. 1500-2000 valence electrons. The spectrum is then calculated using the INDO/CIS method, with the spectroscopic parametrization proposed by Ridley and Zerner. The solvatochromic shifts of pyrimidine in water and of beta-carotene in acetone and isopentane are considered first. These exemplify the situations of a polar molecule in a polar environment and of a non-polar molecule in both polar and non-polar environments. Good agreements with experimental shifts are obtained in all cases. In the case of pyrimidine we analyze the relative importance of the different solvation shells and the role of the electrostatic embedding. Results are obtained using explicit solvent molecules with and without an electrostatic embedding. In the first case including the outer solvation shells increases the calculated shift, whereas in the latter it decreases. The solvatochromic shift of beta-carotene is a persistent and difficult problem because the spectrum involves a π-π*excitation between two states of zero dipole moment. The red shift of this transition is obtained both for isopentane and acetone. Finally, we have considered the absorption spectrum of acetone in supercritical water. The characteristic n−π* transition is calculated to suffer a blue shift of 670 cm-1 compared to the gas phase. This is in excellent agreement with the experimental result that places

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this solvatochromic shift in the interval 500-700 cm-1. Analysis of the hydrogen bonds between the solute acetone and supercritical water indicates that 44% of hydrogen bonds persist compared to water in normal thermodynamic condition. This number correlates with the density of supercritical water considered here where the density is 46% of that in normal water. Compared to normal water the n−π* transition of acetone is calculated to suffer a red shift of 730 cm-1. The success of the present approach to study solvatochromic shifts of organic molecules in solution corroborates the importance of the combined use of quantum mechanics and statistical mechanics and exemplifies the usefulness of the semi-empirical method employed.

Acknowledgments We thank Dr. W. R. Rocha, Dr. H. C. Georg and Dr. D. Trzesniak for discussions and collaboration. We also thank PhD candidate Rafael C. Barreto for the illustration shown in Figure 4. The work reported here has been partially supported by CNPq, CAPES and FAPESP (Brazil).

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