The ligand field spectrum of the hexafluorochromate

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An ab initio study including correlation effects. K. Pierloot and L. G. ... Further improvement of the 4A 2g -+ 4T2g transition was obtained by extending the CI ... ferent crystal environments.24-27 In all cases, the authors report three broad ...
The ligand field spectrum of the hexafluorochromate (III) anion: An ab initio study including correlation effects K. Pierloot and L. G. Vanquickenborne a ) Department of Chemistry, Uniuersity of Leuuen, Celestijnenlaan 200F, B-3030 Heuerlee-Leuuen, Belgium

(Received 23 March 1990; accepted 21 May 1990) The ground and ligand field excited states of CrF~ - have been studied, using different Gaussian basis sets of atomic natural orbitals. Each state was first optimized separately in a complete active space self-consistent field (CASSCF) calculation, including three active electrons in the 2t2g and 4eg shells. Further correlation was then added by using either a singles and doubles configuration interaction approach (SDCI) or by the average coupled pair functional method (ACPF) with the CASSCF configuration space as the reference space. Thereby the number of correlated electrons was raised up to 15. It is shown that the quartetquartet transitions, corresponding to a 2t2g -+ 4eg excitation, are described already very accurately at the CASSCF level. Further improvement of the 4A 2g -+ 4T2g transition was obtained by extending the CI treatment so as to include the F 2p electrons from the It 2g ,3eg , and finally also from the 6a Ig shell. For the intraconfigurational t ~g quartet-doublet transitions on the other hand, the best results were obtained by an 11 electron CI treatment, including the CrOll) 3s and 3p electrons.

I. INTRODUCTION

The accurate calculation of the electronic structure of transition metal complexes still remains a major challenge to quantum chemistry. Due to the size of these systems, up to now, most calculations have been performed at the SCF or limited CI level. 1-17 Calculations of this type have been able to offer some very valuable qualitative information regarding the metal ligand bonding phenomenon. Yet, if one wants to obtain more accurate results, it becomes necessary to include a more rigorous treatment of electron correlation. 18,21 Calculations on small transition metal systems for example have indicated that very elaborate wave functions are often necessary in order to obtain a quantitatively correct description of spectral properties,22 Thus in a recent CPF study on the ligand field states and lowest charge transfer states of CUF2 and CuC1 2,23 a 25 electron correlation treatment in a very large Gaussian basis set (including up to g-type functions) turned out to be necessary in order to obtain results for the energy separations which are close to chemical accuracy, In this study 8 electrons per ligand (e.g., the 2s and 2p electrons on F or the 3s and 3p electrons on CI) were included in the correlation treatment. It is clear that a similar procedure cannot be maintained if one wants to study larger transition metal systems, including more than two ligands. Instead, a more restrictive choice has to be made as to which electrons should be correlated. In this paper we want to report the results of a detailed study of the ligand field spectrum of the CrF! - anion, using such a restrictive procedure. CrF~ - has an octahedral geometry; in the 4A 2g ground state the three CrOll) 3d electrons are occupying the 2t2g shell. Table I shows the results of several spectroscopic investigations of this complex, carried out in a number of different crystal environments. 24-27 In all cases, the authors report three broad parity-forbidden spin-allowed quarteta)

To whom all correspondence should be addressed.

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J. Chem. Phys. 93 (6), 15 September 1990

quartet bands, corresponding to a single or a double excitation from the 1T type 2t2g shell to the (J' type 4eg shell: 4A 2g -+ 4T2g ,4A 2g -+ 4TIg (a) and 4A 2g -+ 4TIg (b). In addition, three spin-forbidden quartet-doublet bands, corresponding to intraconfigurational 2t ~g transitions, can be observed in the spectra. All three are situated near to one of the quartetquartet bands, from which they borrow their intensity via spin-orbit coupling: 4A 2g -+ 2E g and 4A2g-+2TIg from 4A 2g -+4T2g and 4A 2g -+2T2g from 4A2g-+4TIg(a). As a matter of fact, the mixing of these doublet and quartet states by spin-orbit coupling is quite large in the considered fluoride complex; indeed the magnitude of the ligand field splitting

TABLE I. Spectroscopic data for the hexaftuorochromate (III) anion: a confrontation between the experimental ligand field spectra of a number of different crystals and the results of an open-shell Roothaan Hartree-Fock calculation on Cr F~ - taken from Pierloot et al. (Ref. 17). Band position (cm -

SCF res ults"

I )

Ref. 24" Ref. 25' Ref. 26d Ref. 27' 14900 15700 16400 22700 22000 34400

15060 15670 22780 35100

15200 16300 16300 21800 23000 35000

15600 15040 16600

Assignment 4A,g _ 4T,.

4A

_ 2g

2

E g

4A2g_2Tlg

4A'g _ 4T,g (a) 4A2g_2T2g

4A,s _ 4T,s (b)

(Ref. 17) 13790 22070 22050 26774 36433 30408

The Hartree-Fock Roothaan calculations in Ref. 17 were performed with a (15s lip 6d /9s 6p 4d) basis set on Cr and a (9s 5p/5s 3p) basis set on F. The Cr-F distance was kept at 1.93 A (Ref. 35), which is also the value in the present study. b No countercation reported. c (NH 4 )]CrFo' dK]CrFo· eK,NaCrFo'

a

0021-9606/90/184154-10$03.00

® 1990 American Institute of Physics

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K. Pierloot and L. G. Vanquickenborne: Hexafluorochromate (III) anion

happens to shift the quartets very close to the doublet region. This results in a rather complex band structure of the absorption spectrum, which makes it difficult to calculate the exact separation of the different electronic states. The numbers in Table I indicate energy shifts of the order of 5001000 cm - I for the different transitions from one crystal to another. This results in a varying relative position of the 2Eg and 4T2g state in the first band and of the 2T2g and 4T2g (a) state in the second absorption band. If one wants to calculate these details of the spectra, it is of course necessary to include both the crystal surroundings and the effect of spin-orbit coupling. A study along this line was published by Pueyo and Richardson,6 including a comparison of the spectrum of K 2NaCrF6 and CrF3 • In their study, correlation was included by adding a semiempirical correlation energy correction (CEC). In the present study on the contrary, we will emphasize the effect of correlation on the energy levels of CrF! - . Our calculations will be carried out for the CrF! - ion, without any counterions; the effects of spin-orbit coupling will be neglected throughout. 29 This implies of course that we will have to concede an inherent uncertainty of at least 500-1000 cm - I in our results. However, the main purpose of our study is not to obtain a quantitative reproduction of the experimental spectrum. Rather, we will try to gain an understanding of the correlation phenomenon in CrF~ - and other related complexes. We will start from the HF situation, and we will gradually increase the number of configurations included in the calculation. The last column of Table I shows the result of a Roothaan open-shell HF calculation on the different states of the CrF~ - anion, performed with a (segmented) Gaussian basis set. 17 The numbers in this column are representative of the general trend observed in SCF calculations on the ligand field spectra of octahedral transition metal complexes: I - 1O the open-shell repUlsion energy, as represented by the intraconfigurational2t ~g transitions, is calculated too high, while the ligand field strength lODq, corresponding to the transition 4A 2g -+ 4T2g , is calculated too low. Both calculated 4A 2g -+ 4T Ig transitions in the last column of Table I show a large deviation from the experimental excitation energies: the first transition 4A 2g -+ 4TIg (a), corresponding to a single excitation from 2tg to 4eg, is calculated too high, while the second transition 4A 2g -+ 4T2g (b), corresponding to a double excitation, is calculated too low. As we have shown in a separate paper,17 the result for these two transitions can be improved drastically by a limited CI calculation within the Cr (III) 3d 3 manifold. In the present study we will start from a CASSCF calculation for each of the ligand field states, including three active electrons in the 2t2g and 4eg shells. The resulting molecular orbitals are then used in a multi reference SDCI calculation, in which the entire CASSCF configuration space is used as the reference space. Calculations correlating up to 15 electrons have been carried out. The size of the CI matrices varied between 28 000 and 555000, depending on the basis and the symmetry. In all calculations where more than three electrons were correlated, the effect of quadruple excitations was incorporated, either by adding the multi-

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reference analog of the Davidson correction,31.32 or by replacing the SDCI calculation by an ACPF treatment. 33.34

II. CALCULATIONAL DETAILS

All calculations were carried out with the Lund University quantum chemistry software MOLCAS,36 using Gaussian ANO type basis sets?7 Both for the Cr(lll) ion and on the F - ligands, two different basis sets were considered. For Cr(lIl) we started with a (16s 12p 8d)/[7s 5p 3d] set, while in the second set four primitive! functions were added, contracted to two functions. 38 The smallest F basis set is ( 14s 9p) / [5s 3p]. Here too, this set is further extended in a second basis set, with four d polarization functions contracted to one. 39 The calculations were performed with the following three combinations of basis sets: Basis 1: (16s 12p 8d)/[7s 5p 3d] on Cr and (14s 9p)/ [5s3p]onF, Basis 2: (16s 12p 8d 4j)/[7s 5p 3d 2j] on Cr and (14s 9p)/[5s 3p] on F, Basis 3: (16s 12p 8d 4j)/[7s 5p 3d 2j]on Cr and (14s 9p 4d)/[5s 3p Id] on F. In all calculations, only the pure spherical harmonic components of the basis functions were used. The symmetry group used in the calculations is D 2h , although the geometry of the complex was taken to be perfectly octahedral; we will therefore use the Oh notation for all representations. Since an accurate determination of the bond distance is not possible without taking into account the crystal surrounding of the complex, we decided to perform all calculations at a fixed Cr-F distance of 1.93 A, the experimental bond distance of the K2NaCrF6 complex. 35 In order to understand the effects of correlation in CrF! -, we carried out some preliminary calculations on Cr(lll). In the CASSCF treatment ofthis atomic ion, symmetry restrictions were imposed within each D2h representation, in order to prevent mixing of orbitals with a different I quantum number. III. RESULTS AND DISCUSSION A. Results for Cr(lII)

In ligand field terms, CrF! - is known to be a "weak field" complex. This means that the sequence of the ligand field states in the complex closely resembles the sequence of the parent CrOll) free ion terms. Experimentally, the following energy sequence is found for the different states corresponding to the Cr( III) 3d 3 configuration: 40 (1)

In an octahedral environment, the lowest terms in this sequence are split as follows: 4 4F-+ A 2g +4T2g +4TIg , 4p-+4T 2G

-+

Ig

2E g

,

+ 2TIg + 2T2g + 2A Ig·

(

2)

Therefore, the various transitions of the ligand field spectrum ofCrF! - correspond to only two transitions in the free

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K. Pierloot and L. G. Van quicken borne: Hexafluorochromate (III) anion

TABLE II. Experimental vs calculated energies of the lowest energy terms of Cr (III) a . All calculations were performed with two different ANO basis sets: Basis I = (16s 12p 8d) / [7s 5p 3d]; basis 2 = (16s 12p 8d 4j) / [7s 5p 3d 211. (A) Experimental spectrum (cm 4p 2G

1 )b

13 758 14699

(B) SCF results Basis I

Basis 2

NHF

- 1041.47055

- 1041.47055

- 1041.477 18

17173 17818

17173 17818

17562 17826

Ground state energy (Ha)

4F

Spectrum (cm 4p 2G

1)

(C) CI (3d): 3 electrons correlated Basis I

Basis 2

- 1041.48029

- 1041.48998

16700 17095

15754 17105

Basis I

Basis 2

- 1041.64802

- 1041.75830

14619 16570

14159 16091

- 1041.65337

- 1041.77021

14382 16502

13 673 15834

- 1041.65250

- 1041.76834

14461 16515

13800 15854

Ground state energy (Ha)

4F Spectrum (cm - 1 ) 4p 2G (D) CI(3s + 3p + 3d): II electrons correlated CI(SD) Ground state energy (Ha)

4F

Spectrum (cm 4p 2G

1)

CI(SD) +Q Ground state energy (Ha) 4

Spectrum (cm _. 1 ) 4p 2G ACPF Ground state energy (Ha)

4F

Spectrum (cm 4p 2G

1)

Since all calculations were performed in D2h symmetry (with restrictions imposed as indicated in Sec. II), a slightly different energy was obtained for the different components of a degenerate atomic state. The difference never exceeded 50 cm - 1 , and the table shows the average value. b From Ref. 40; the energies given were obtained by averaging over the different J levels in each term. a

Cr (III) ion. The experimental transition energies are shown in Table II(A). Table II(B) shows the corresponding SCF results, calculated with the two basis sets on chromium (with and without! functions), and compared to the results of a numerical Hartree-Fock calculation. 41 The most striking point of Table II (B) is the fact that the 4F ..... 4p transition is described so poorly at the SCF level, even worse than the 4F ..... 2G transition. Normally, one would expect the differential correlation error between two states originating from the same 3d n configuration to be smaller for two sates with the same spin than for two states with a different spin. In order to improve the result for both transitions, we decided to carry out a single reference CI calcula-

tion for the three states under consideration, starting from the SCF wave function and allowing all single and double excitations. In a first step, only the three 3d electrons were correlated. The result is shown in Table II(C). In a second step, we decided to add the 3s and 3p electrons to the correlation treatment thus ending up with 11 correlated electrons. The results are shown in Table II(D). With this number of correlated electrons, the effect of higher than double excitations may become important. This effect was estimated in two different ways, first by adding a Davidson correction (denoted as + Q), and alternatively by performing an ACPF calculations. From a comparison of the various numbers in Table II, it is clear that including the 3s and 3p

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K. Pierloot and L. G. Vanquickenborne: Hexafluorochromate (III) anion

electrons is crucial if one wants to get a correct description of the differential correlation between the different terms corresponding to the Cr (III) 3d 3 configuration. 42 With the largest basis set, the experimental 4F -+ 4p transition is reproduced quasi exactly. For the 4F -+ 2G transition, an error of about 1200 cm - I still remains. By comparing the results for both basis sets in Table II(D), one can see that adding the two/polarization functions to the basis set leads to a distinct improvement for both the 4 F -+ 4 P and the 4 F -+ 2G transition. This also suggests that the results are not totally converged with respect to the basis set, and that a better description of the 4F -+ 2G transition can probably be obtained by adding more polarization functions (f and also g type). However, since it is our intention to compare the results for the free Cr(lll) ion with the corresponding transition energies in CrF! - , calculated with the same basis set, we decided not to enlarge the basis set on chromium any further. Instead the value of 1200 cm - I will be used as an indication of the error to be expected for quartet-doublet transitions in the spectrum of CrF! - . B. CrF~-: CASSCF

In order to obtain an acceptable reference for a configuration interaction treatment in the CrF~ - ion, we decided to start from a CASSCF calculation with the 3 valence electrons in an active space consisting of the (2t 2g + 4eg ) shells, which are essentially Cr 3d orbitals. In this way it is possible to deal with all near-degeneracy effects within the ligand field states. Due to the high symmetry of the complex, the number of states involved in this MCSCF treatment is rather limited, namely one 4A 2g , one 4T2g , and two 4T1g quartet states, and four 2Eg , five 2T1g , and five 2T2g doublet states.

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This means that for the quartet states the situation is very simple: both 4A 2g and 4T2g are subjected to an SCF treatment, while the introduction of the CAS formalism will push the two 4T1g states apart. This is confirmed by Table III (A), showing the CASSCF results for the three different basis sets. First of all it is clear that the addition of polarization functions to the basis sets does not have a significant effect on the different transition energies at the CASSCF level-the transitions are equally well described with the smallest basis set. From a comparison with the SCF results in Table I, one can furthermore see that the 4T2g - 4A 2g energy separation, which in ligand field terms equals lODq, indeed remains practically unchanged. 44 The two 4T1g terms do show a considerable configurational mixing. The following state composition results from the CASSCF calculation: 4T1g (a):

0.85 4T,g(t~ge~)+0.52 4TIg(t~ge;),

4T,g (b):0.61 4TIg(t~ge~)-0.79 4Tlg(t~ge;).

(3)

The mixing between both 4T1g states in the complex is accompanied by substantial energy shifts: the energy of the 4T1g (a) state is lowered by about 5000cm - I as compared to its SCF value (in Table I), while the energy of the 4T1g (b) state is raised by the same amount. As a remarkable result, the transition energy for both states now almost exactly equals the experimental value. This is the more striking in view of the fact that for an equally good description of the parent 4F -+ 4p transition in the free Cr(lll) ion [corresponding to the 4A 2g -+ T ,g (a) transition in the complex], an 11 electron correlation treatment proved to be necessary. The reason for this disparity is not clear. As compared to the SCF results of Table I, the CAS formalism also leads to a substantially better description of

TABLE III. Calculated spectrum of CrF~ -, obtained by correlating the three valence electrons in the (2t2• + 4e.) shell (predominantly Cr 3d).

(A) CASSCF results Ground state energy (Ha) 4A 2• Spectrum (cm - , ) 4T2• 4T,.(a) 4T,g (b) 2E. 2T,g 2T2• (B) SDCI results Ground state energy (Ha) 4A 2• Spectrum (cm - , ) 4T2g 4T,.(a) 4T,.(b) 2E. 2T,. 2T2g

Basis 1

Basis 2

Basis 3

- 1640.151 83

- 1640.17005

- 1640.17994

13420 21340 35120 19770 20770 27690

13 390 21630 35430 19880 20790 27890

13 430 21690 35510 19820 20800 27900

-1640.16269

-1640.19022

- 1640.200 17

13560 21370 34940 18880 19900 26440

13 550 21380 34600 18840 19870 26010

13 590 21430 34670 18830 19870 26020

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K. Pierloot and L. G. Vanquickenborne: Hexafluorochromate (III) anion

the intraconfigurational t ~g quartet-doublet transitions, due to the mixing with energetically higher 2Eg , 2T1g , and 2 T 2g states (originating from different t ~gconfigurations). The largest improvement is obtained for the 4A 2g ~2T2g transition, which is reduced by about 9000 cm - 1 • However, on the whole, the calculated values for the quartet-doublet transitions still deviate by about 4000-5000 cm - 1 from the corresponding experimental transition energies. As can be seen from Table II, this is comparable to the error obtained at the SCF level for the parent 4F...... 2G transition in Cr(lll). As for the free ion, a further improvement of the transition energies can only be obtained by performing large-scale CI calculations, going beyond the ligand field states.

xe;

C. CrF~-: A 3 electron CI treatment

Table III(B) shows the result of a first attempt in this direction. The CASSCF configuration space now served as the reference for an MRCI calculation, based on the CASSCF orbitals and including all single and double excitations of the three valence electrons. By comparing the corresponding numbers in Table III(A) and III(B) one can see that, while this CI treatment does not seem to have a very pronounced effect on the quartet-quartet transitions, the quartet-doublet transitions are approaching the experimental energies by an amount of about 1000-2000 cm - I . With the exception of the 4A 2g ...... 2T2g transition, which seems to get a slightly better description when two f functions are added to the Cr(lll) basis set, enlarging the basis sets with polarization functions again does not alter the results to any significant extent. Obviously there is not much use in using extended basis sets when only a limited number of electrons are correlated. Overall, it is clear that, even with a 3 electron correlation treatment in a nonextended basis set the experimental spectrum is reproduced rather accurately. In order to improve the agreement between theory and experiment even further, it is necessary to include more electrons in the CI treatment. Now, if we exclude all the chromium Is, 2s, 2p and all the F Is core electrons, we are still left with a total of 59 valence electrons that might be of potential significance in the correlation problem. Therefore, inevitably, we will have to make a selection. The most obvious choice would probably be to take the electrons from the highest lying valence orbitals. In Table IV (A) we have collected the valence orbitals that result from the CASSCF calculation on the 4A 2g ground state, in increasing order of energy. Table IV(B) shows the result of a Mulliken population analysis on the same orbitals. From the populations in Table IV(B) it is clear that CrF~ - is a very ionic complex. This also implies that every molecular orbital can be characterized as being either an almost purely chromium or an almost purely fluorine orbital. The main character of each orbital is also shown in Table IV(A). As one can see, orbitals with the same atomic orbital parentage are grouped together in the diagram, in the following order: Cr(3s) < Cr(3p) < F(2s) < F(2p) < Cr(3d). The Cr 3s and 3p orbitals which were shown (Sec. III a) to

TABLE IV. Orbital energies (A) and population analysis (B) of the 4A,. ground state of CrF! - (CASSCF results).' (A) Orbital energies Orbital

Energy (Ha)

4a'g 3t," Sa,. 2eg 4t," 6a,g It,. 3e. 5t," It," 6t," It,.

- 3.0411 - 1.7887 - 0.9427 - 0.9267 - 0.9247 - 0.0874 - 0.0632 - 0.0598 - 0.0562 -0.0171 - 0.0092

2~

Q3TI8

Cr(3s) Cr(3p) F(2s) F(2s) F(2s)

F(2Pa) F(2pff) F(2p,,) F(2p,,;2prr) F(2prr) F(2p,,;2Prr) F(2Prr) Cr(3d rr )

0.0022

(B) Population analysis Cr

P

6.19 12.21

F

s Pa Prr

dO" d rr Total Charge U

0.53 3.18 22.11 + 1.89

Main character

Total Charge

3.97 1.88 3.96

9.81 -0.81

All numbers given in this table result from a CASSCF calculation on the 4A,. ground state, using basis 1. The addition of polarization functions to the basis set slightly alters the population analysis results, in a predictable direction: the addition of/polarization functions on Cr (basis 2) induces an increase of the electronic population on this ion, resulting in 22.72 e -on Cr and 9.71 e - on F; the addition of the d function on F (basis 3) has a compensating effect, so that we now end up with 22.35 e - on Cr and 9.77 eo. on F.

play an important role in the correlation problem of the Cr(lll) free ion, are now situated below all the ligand valence orbitals. Yet it is clear that the same orbitals are likely to be important when considering the intraconfigurational 2t ~g quartet-doublet transitions in the complex. Therefore, in Sec. (III D), we will skip the ligand orbitals altogether, and consider an 11 electron correlation treatment, involving the electrons from 4a lg , 3t 1u ' and 2t2g , which are predominantly chromium 3s, 3p, and 3d atomic orbitals. Finally in Sec. III E we will report the results of a set of calculations, where the correlation problem will be further refined by including a number of orbitals with predominant F 2p character. D. CrF~-: An 11 electron correlation treatment

As for the 3 electron MRCI treatment of the previous section, the CI calculation is based on the orbitals that result from the CASSCF calculation for the state under consideration, and all the ligand field states are included in the reference space. The contributions to the correlation energy from unlinked cluster terms have-ueen a"CCounted for by applying the multireference analog of the Davidson correction (denoted as + Q) on the one hand, and alternatively by per-

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K. Pierloot and L. G. Vanquickenborne: Hexafluorochromate (III) anion

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TABLE V. Results of the 11 electron correlation treatment, involving the electrons from the orbitals 4a lg (predominantly Cr 3s), 3t 1u (predominantly Cr 3p), and (2t2• + 4e.) (predominantly Cr 3d).

(A) MR CI(SD) Ground state energy (Ha) 4A 2• Spectrum (cm - 1 ) 4T,. 4T1.(a) 4T1g (b) 'Eg 'T,g 'T,. (B) MR CI(SD) + Q Ground state energy (Ha) 4A,. Spectrum (cm - 1 ) 4T,g 4T1.(a) 4T1g (b) 'Eg 'T,. 'T,g

(C) ACPF Ground state energy (Ha) 4A 2• Spectrum (cm - , ) 4T'g 4T,.(a) 4T1g (b) 2Eg 'T1g 2T'g

Basis 1

Basis 2

Basis 3

- 1640.32401

- 1640.44931

- 1640.460 10

13 370 20810 33090

13 270 20600 32670

13 330 20690 32890

18120 19420 24430

17690 18860 24750

17690 18870 24680

- 1640.32888

- 1640.460 34

- 1640.471 18

13360 20680 32630

13 260 20350 31560

13320 20430 31680

17640 19090 23560

17250 18550 24100

17260 18560 24120

- 1640.32808

- 1640.45863

- 1640.46947

13360 20750 32920 18000 19330 24210

13 260 20530 32530 17460 18670 24420

13 330 20610 32 660 17 470 18680 24440

forming an ACPF calculation. The results are shown in Table V. As was to be expected from the results for Cr(lll) (Table II), the inclusion of the Cr 3s and 3p electrons in the correlation treatment has a favorable effect on the 1 ~g intraconfigurational quartet-doublet transition energies in CrF! - . For all three transitions an improvement of about 1500 cm - 1 is obtained in comparison with the results of the 3 electron CI treatment [Table III(B)], at least when considering the numbers obtained with basis set 2 or 3. In the same way as for the free ion, the two/polarization functions on Cr (III) now also start to playa distinct role, and a further improvement of the quartet-doublet transitions could probably be obtained by using an even larger basis set on this ion. The d polarization functions on the ligand orbitals on the other hand, added in basis set 3, are not able to lead to any further improvement: the numbers from the second and third column of Table V(B) differ by at most 50 cm - I . This is not a very surprising result: due to the ionic nature of

the complex, the intraconfigurational 21 ~g transitions essentially take place in the metal 3d shell. The table also shows that the effect of triple, quadruple, ... excitations on the transition energies appears to be rather limited. The Davidson correction produces even slightly better results than ACPF, although the difference never exceeds 300 cm - 1 • The good agreement between both methods may serve as an indication that the effect of higher excitations has properly been accounted for in both correction schemes.

E. CrF:-: Correlation treatment including electrons from the F(2p) orbitals For the quartet-quartet transitions, the results of Table V are less encouraging: the 4A 2g -+ 4T2g result does not improve as compared to the 3 electron treatment of Sec. III C, while both of the other transitions 4A 2g -+ 4T1g (a) and 4A 2g -+ 4T1g (b), which were reproduced quasi exactly at the 3 electron correlation level, become considerably worse. Ap-

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K. Pierloot and L. G. Vanquickenborne: Hexafluorochromate (III) anion

parently the insertion of the Cr(Ill) and 3s and 3p electrons in the correlation treatment does not necessarily lead to an improvement if one wants to describe transitions in which the ligand field is more directly involved. Even in a very ionic complex like CrF~ - , there is still a slight difference in metalligand mixing between u and 1T bonding types (differential covalency),17 the u bonding in general being slightly more covalent. This is illustrated by the population analysis results in Table IV. Both the populations on the central Cr(Ill) ion and on the F ligands are indicative of a larger ligand-to-metal charge donation through the u bond than through the 1T bond: as compared to free F - , the population of 2Pa on each ligand is decreased by 0.12 electrons, while only 0.02 electrons have left each 2p" orbital. The central chromium ion on the other hand receives-when compared to free Cr3 + -0.53 electrons in its 3d a and only 0.18 electrons in its 3d" orbital. In view of these considerations it is reasonable to suppose that if we want to obtain results for the quartet-quartet transitions which are more accurate than those obtained in the 3 electron correlation treatment (Table III), we will in

addition have to correlate the electrons in the molecular orbitals 5a ,g up to It ,g (see Table IV), which have mainly ligand character. If, in addition to the II Cr(Ill) valence electrons, all electrons in these ligand orbitals were included in the correlation treatment, we would undoubtedly be able to compensate for the error made in the 4A 2g ..... 4T1g transitions at the II electron correlation level (see Table V). However, such a calculation-if at all possible-would require a huge amount of computer time. Instead, we decided in favor of a more economic solution, by removing the chromium 3s and 3p from the correlation treatment, and including instead a limited number of electrons from the MO's with F 2p character. The most obvious candidates to consider for such a limited CI treatment are of course the orbitals It 2g and 3eg , which are the ligand counterparts of the metal-like orbitals involved in the quartet-quartet transitions. When adding the electrons in these orbitals to the three chromium 3d electrons, we end up with a 13 electron correlation problem, which is perfectly manageable, even with the largest basis set (the number of included configurations never exceeded 555000). The results are shown in Table VI. As in the pre-

TABLE VI. Results from the 13 electron correlation treatment, involving the electrons from the orbitals (112 • + 3eg ) (predominantly F 2p) and (212 , + 4e.) (predominantly Cr 3d).

(A) CI(SD) Ground state energy (Ha) 4A 2• Spectrum (cm - I ) 4T2• 4T,g (a) 4T,s (b)

(B) CI(SD) + Q Ground state energy (Ha) 4A 2• Spectrum (cm - I ) 4T2• 4T,g (a) 4T (b)

'g

2Eg

2T,• 2T 2g

(C) ACPF Ground state energy (Ha) 4A 2g Spectrum (cm - I ) 4T2• 4T,.(a) 4T,• (b)

Basis 1

Basis 2

Basis 3

- 1640.22651

- 1640.26083

- 1640.29731

14220 21950 35100

14290 22040 35070

14220 21950 34920

18710 19620 26190

18600 19530 25700

18620 19560 25740

- 1640.22892

- 1640.26401

- 1640.301 61

14310 21970 34940

14400 21960 34710

14330 21860 34570

18650 19540 26030

18490 19400 25430

18500 19410 25430

41640.22867

- 1640.263 73

- 1640.301 22

14300 22000 35050

14390 22080 34880

14330 22000 34850

18660 19560 26100

18520 19440 25550

18540 19460 25570

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K. Pierloot and L. G. Vanquickenborne: Hexafluorochromate (III) anion

vious section, all ligand field states were included in the reference space, and the effect of unlinked cluster terms was estimated by means of a Davidson correction (denoted by + Q) as well as by performing an ACPF calculation. The results for the transition energies in Table VI have to be compared with the corresponding results from the 3 electron correlation treatment of Table III(B). In doing so, the following points should be noted: (i) The quartet-doublet transitions are hardly better described at the 13 electron correlation level: the improvement never exceeds 600 cm - I . This confirms the point made in the previous section, that not the ligand orbitals, but the Cr 3s and 3p orbitals are crucial in the description of the differential correlation between the different terms of the t ~g ground state configuration. It also suggests that the quartetdoublet transition energies of Table V could be further improved by the roughly same amount, about 400-600 cm - I , in a combined 21 electron treatment, involving all the MO's considered thus far. (ii) The results for the quartet-quartet transitions clearly indicate that including the It 2g and 3e g shell in the correlation treatment is indeed a good starting point if one wants to describe ligand field transitions between terms with the same spin. The transition energies of both 4A 2g -+ 4TIg transitions are now definitely consolidated at about 22 000 and 35 000 cm - I , in perfect agreement with the experimental values of Table I. For the 4A 2g -+ 4T2g transition the result is also very satisfactory: by correlating the 13 electrons under consideration the differential correlation error between both quartet states has been reduced by half of its value in Table V. The resulting value of lODq now approaches the experimental value to about 1000 cm - I .

4161

(iii) The transition energies in Table VI are remarkably indifferent to any change in the basis sets. This was also the case at the 3 electron correlation level (Table III), and should probably be considered as an indication of the ionicity of the complex under consideration, involving very little mixing between the metal and ligand orbitals. (iv) The triple, quadruple, ... excitations from the concerned orbitals apparently do not contribute very much to the differential correlation between the ligand field states. The results from the SDCI, SDCI + Q treatment are quite conform; the difference between them never exceeds 4oocm- l . In a certain sense, the results of Table VI are a confirmation of the ligand field idea, describing the bonding between the central metal ion and its surrounding ligands exclusively with the metal d orbitals. The validity of this theory in Werner-type complexes like CrF! - had already been confirmed on several occasions at the ab initio SCF and limited CI level. 15, 17 Yet the calculated values of the ligand field strength were almost inevitably too small in comparison with the experimental value. The present calculations show that a substantial part ofthe error can be removed by correlating the electrons in the d type t 2g and eg orbitals, as well as in their ligand counterparts of the same symmetry. Basically, this is still a d-only description of the metal-ligand bonding, where the 4s and 4p metal orbitals are only of secondary importance. In order to verify this conclusion, we decided to perform a final calculation on the quartet-quartet transitions, in which we now also included the electrons from the 6a lg orbital. In view of the above considerations, the result of this calculation should serve as an indication for the involvement

TABLE VII. Results from the IS electron correlation treatment, involving the electrons from the orbitals (6a,g + 1t2• + 3eg ) (predominantly F 2p) and (2t 2• + 4eg ) (predominantly Cr 3d).

(A) CI(SD) Ground state energy (Ha) 4A 2 • Spectrum (cm - , ) 4T

2g

4T'g(a) 4T'g (b)

(B) CI(SD) + Q Ground state energy (Ha) 4A 2g Spectrum (cm .,) 4T

2g

4T'g (a) 4T'g(b)

(C) ACPF Ground state energy (Ha) 4A 2g Spectrum (cm - , ) 4T2g 4T,g(a) 4T,.(b)

Basis 1

Basis 2

Basis 3

- 1640.24955

- 1640.285 77

- 1640.33355

14350 22000 35290

14420 21960 35150

14300 21790 35070

- 1640.25342

- 1640.29074

- 1640.340 46

14470 22010 35080

14560 21960 34730

14460 21790 34620

- 1640.253 15

- 1640.29045

- 1640.34009

14470 22070 35240

14560 22140 35090

14470 21980 35010

J. Chern. Phys., Vol. 93, No.6, 15 September 1990 Downloaded 23 May 2013 to 143.106.51.149. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

4162

K. Pierloot and L. G. Vanquickenborne: Hexafluorochromate (III) anion

of the Cr 4s orbitalin the metal ligand bonding. The resulting transition energies are shown in Table VII. The results in this table are in line with the expectations: as compared to Table VI no significant shift of the 4A 2g -+ 4TIg transitions can be detected. The lODq transition 4A 2g -+ 4T2g is raised very slightly, by about 150 cm - I. This result is an unambiguous indication that the Cr 4s orbital is not strongly involved in the Cr-F bonding. IV. CONCLUSION

The present investigation on the ligand field spectrum of CrF! - has demonstrated that it is feasible to devise accurate calculations on the spectral properties of relatively large transition metal systems. No extravagantly large basis sets and a rather limited number of correlated electrons were necessary in order to obtain a quantitatively correct description of the quartet-quartet transitions. The quartet-doublet transitions could probably be further improved by adding more polarization functions on the central Cr( III) ion. Yet, even with only two f functions on this ion the computed quartet-doublet separations are accurate to 1000--2000 em-I.

Also noteworthy, concerning the size of the basis sets, is the fact that at all calculational levels the ligand field separations proved to be completely indifferent to the addition of polarization functions on the F ligands. This is probably due to the ionic character of the Cr-F bonding, and is in a sense encouraging, since it indicates that no extended basis sets will be needed in order to calculate the electronic spectrum of other ionic Werner-type complexes. The present work on CrF~ - suggest that the result of such calculations would at least be accurate enough to be able to aid in the interpretation of experimental spectroscopic studies. The results will probably be slightly modified by including the externallattice potential; the results of the CrF! - calculations imbedded in the appropriate Madelung potential will be published in a separate communication. Finally it should be noted that the complex under consideration-and in fact all octahedral d 2 and d 3 and low-spin d 4 complexes-are rather special, in that a clear distinction can be made in these complexes between the spin-Jorbidden intraconfigurational t ~g transitions on the one hand and the spin-allowed intraconfigurational t 2g ..... eg transitions on the other hand. As we have shown in the present study, the orbitals which are crucial in a limited correlation treatment are different for both types of transitions. In systems with more electrons in the d shell, a similar distinction can no longer be made. This implies that for the description of their electronic spectra a larger correlation treatment, combining the metal 3s and 3p shell and the ligand 2p shell, will probably be necessary. ACKNOWLEDGMENTS

The authors are indebted to the Belgian National Fund for Scientific Research and to the Belgian Government (Programmatie van het Wetenschapsbeleid) for financial support. We thank Dr. B. Roos (University of Lund) and

Dr. P.-O. Widmark (IBM Sweden) for their stimulating help in the use of the MOLCAS program system.

I J. W. Richardson, D. M. Vaught, T. F. Saules, and R. R. Powell, J. Chern. Phys. 50, 3633 (1969). 2H. M. Gladney and A. Veillard, Phys. Rev. 180,385 (1969). 3A. J. H. Wachters and W. C. Nieuwpoort, Phys. Rev. B 5, 4291 (1972). 4J. Demuynck and A. Veillard, Theor. Chim. Acta (Berlin) 28, 241 (1973). 5J. Demuynck, A. Veillard, and U. Wahlgren, J. Am. Chern. Soc. 95, 5563 (1973). 6J. W. PueyoandJ. W. Richardson,J. Chern. Phys. 67,3577,3583 (1977). 7T. J. M. Smit, W. C. Haas, and W. C. Nieuwpoort, Theor. Chim. Acta (Berlin) 43, 277 (1977). • E. Miyoshi, S. Obara, T. Takada, H. Kashiwagi, and K. Ohno, Int. J. Quantum Chern. 19,481 (1981); 23,1753 (1983). 9S. Gutierrez Orellana and L. Pueyo, Solid State Chern. 55, 30 ( 1984). IOL. G. Vanquickenborne, L. HaspesJagh, M. Hendrickx, and J. Verhulst, Inorg. Chern. 23,1677 (1984). II C. W. Bauschlicher and P. S. Bagus, J. Chern. Phys. 81, 5889 ( 1984). 12M. Florez, L. Seijo, and L. Pueyo, Phys. Rev. B 34,1200 (1986). 13 R. Broer, G. Aissing, and W. C. Nieuwpoort, Int. J. Quantum Chern. 29, 1059 (1986). 14 P. Carsky and A. Dedieu, Chern. Phys. 103, 265 (1986). 15 L. G. Vanquickenborne, M. Hendrickx, D. Postelmans, I. Hyla-Kryspin, and K. Pierioot, Inorg. Chern. 27, 900 (1988). 16K. Pierloot, J. Verhulst, P.Verbeke, and L. G. Vanquickenborne, Inorg. Chern. 28, 3059 (1989). 17 K. Pierioot, J. Vandersmissen, and L. G. Van quicken borne (submitted). 18S. Larsson, B. O. Roos, and P. E. M. Siegbahn, Chern. Phys. Lett. 96, 436 (1983). I·H. Johansen and N. K. Andersen, Mol. Phys. 58, 965 (1986). 20 H. Johansen, in Understanding Molecular Properties, edited by J. A very (Reidel, Dordrecht, 1987) p. 95. 21 A. Veillard, A. Strich, C. Daniel, and P. E. M. Siegbahn, Chern. Phys. Lett. 141,329 (1987). 22 S. R. Langhoff, C. W. Bauschlicher, and J. Partridge, J. Chern. Phys. 89, 396 (1988). 23C. W. Bauschlicher and B. O. Roos, J. Chern. Phys. 91, 4785 (1989). 24 C. K. Jorgensen, Absorption Spectra and Chemical Bonding in Complexes, (Pergamon, London, 1962). 25H. L. Schlafer, H. Gaussman, and H. U. Zander, Inorg. Chern. 6, 1528 (1967). 26G. C. Allen, G. A. M. EI-Sharkawy, and K. D. Warren, Inorg. Chern. 10, 2538 (1971). 27 J. Ferguson, H. J. Guggenheim, and D. L. Wood, J. Chern. Phys. 54, 504 (1971); L. Dubicki, J. Ferguson, and B. Van Oosterhout, J. Phys. C 13, 2791 (1980). 28 J. Ferguson, K. Knox, and D. L. Wood, J. Chern. Phys. 35, 2236 (1961); 37,193 (1962). 29 It is true that spin-orbit coupling is rather important in the spectra of CrF~ - and other Cr(lll) complexes, in that it modifies the intensities of the spin-forbidden lines to a significant extent. Yet, its energetic effect on the transitions can be expected to be only of minor importance. For the free Cr(lll) ion equals 273 cm - I (Ref. 30). In CrF~ -, this value is reduced to approximately 170 cm - I (Ref. 28). 30 J. S. Griffith, The Theory o/Transition Metal Ions (Cambridge University, Cambridge 1961). 31 S. R. Langhoffand E. R. Davidson, Int. J. Quantum Chern. 8, 61 (1974). 32M. R. A. Blomberg and P. E. M. Siegbahn, J. Chern. Phys. 78, 5682 (1983). 33 R. Ahlrichs, P. Scharf, and C. Ehrhardt, J. Chern. Phys. 82, 890 (1985). 34R. J. Gdanitz and R. Ahlrichs, Chern. Phys. Lett. 143,413 (1988). 35 S. Schneider and R. Z. Hoppe, Anorg. Allg. Chern. 376, 268 (1970). 36MOLCAS is an electronic structure software written by J. Almlof, M. Blomberg, G. Karistrom, P.-A.. Malmqvist, B. O. Roos, A. J. Sad1ej, P. E. M. Siegbahn, and P.-O. Widmark. 37J. Almlofand P. R. Taylor, J. Chern. Phys. 86, 4070 (1987). 3·P._O. Widmark (unl?ublished results). 39 P.-O. Widmark, P.-A. Malmqvist, and B. O. Roos, Theor. Chim. Acta. (in press).

t

J. Chern. Phys., Vol. 93, No.6, 15 September 1990 Downloaded 23 May 2013 to 143.106.51.149. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

K. Pierloot and L. G. Vanquickenborne: Hexafluorochromate (III) anion

J. Sugar and C. Corliss, J. Phys. Chern. Ref. Data 6, 317 (1977). C. H. Fischer, The Hartree-Fock Method for Atoms (Wiley Interscience; New York, 1977). 42 In Ref. 43 it was shown already that the exclusion of the 3s and 3p electrons from the correlation treatment may result in errors of a few tenths of an eV in the computed excitation energies in first-row transition metal atoms. 43M. Pelissier and E. R. Davidson, Int. J. Quantum Chern. 25, 483 (1984). 44 Apart for a small basis set effect, the difference between the calculated value for the'A 2..... 4T2 • transition energy in Table I (ROHF) and Table

40 41

4163

IV (CASSCF) is due to the fact that both calculations were performed using a different symmetry point group. The ROHF calculations were performed using the full symmetry of the complex, 0h. Thereby the different components of the e. and t 2 • representations were forced to be degenerate. This degeneracy is, however, no longer imposed in the D2h symmetry group, which was used in the CASSCF calculation. This results in an increase of the degrees of freedom for the degenerate 4T2g state, and thus to a (limited) lowering of the energy of this state with respect to the nondegenerate 'A 2• ground state.

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