Ab initio and density functional study of the Jahn-Teller distortion in the silane radical cation

29 November 1996

CHEMICAL PHYSICS LETTERS

ELSEVIER

Chemical Physics Letters 262 (1996) 782-788

Ab initio and density functional study of the Jahn-Teller distortion in the silane radical cation F r a n k D e Proft, P a u l G e e r l i n g s * Eenheid Algemene Chemie, Vrije Universiteit Brussel, Faculteit Wetenschappen, Pleinlaan 2, B-1050 Brussels, Belgium Received 23 September 1996

Abstract

An ab initio and density functional study is made of the Jahn-Teller distortion of the silane radical cation using Dunning's cc-pVTZ basis set. For this cation, the D2d, C3v, C2v and Cs structures are fully optimized and analyzed using an energy component analysis. The distortion towards the lowest energy geometry results in an increase in kinetic energy and a decrease in the electron-nuclear attraction energy, electron-electron repulsion energy and nuclear-nuclear repulsion. The exact exchange density functional methods B3LYP and B3PW91 turn out to be the methods best suited for the study of geometries, energies and vibrational spectra of these cations.

1. I n t r o d u c t i o n

According to the Jahn-Teller theorem [1-3], a symmetrical non-linear molecule with a partially filled and degenerate highest occupied molecular orbital will be distorted such that the degeneracy will be raised. Radical cations of the tetrahedral species XH4 undergo Jahn-Teller distortion from their initial Td symmetry. The symmetry lowering of a such a tetrahedral system usually results in species having C3v, D2~ or C2v symmetries. For CH +, it was shown, both from experiment and high level ab initio calculations, that the ground state is a C2v strucure (a 2Bl electronic state) (see Ref. [4-6] and references therein). Boyd et al. [6] studied the methane radical cation using an energy component analysis. It was shown that the Jahn-Teller distortion leads to a decrease of the expectation value of the electronnuclear attraction energy, an increase in the expec* Corresponding author.

tation value of the interelectronic repulsion energy and an increase in the internuclear repulsion energy. The interelectronic repulsion was found to play a dominant role in determining the relative energies of the possible Jahn-Teller distorted structures; electron correlation effects were found to be relatively unimportant. For Sill +, only theoretical studies have been presented hitherto [ 7-9,5,10-12]. At the HF/4-31G level, Gordon found the C3v structure (2Ai state) to be 10.9 kcal/mole below the C2v structure and 23.7 kcal/mole below the D2o structure (2B2 state) [7]. Caballol, Catala and Problet [9] studied Jahn-Teller distortions in SiH~-, GeH + and SnH~- using CI level calculations and pseudopotentials. In this case, it was concluded that the C2v structures were the lowest in energy in all cases, although the energy gap C2v-C3v decreased when going from SiH~- to SnH~-. They also concluded that the inclusion of correlation effects was important for an accurate determination of the geometrical parameters and energy differences of these species. At the HF/6-31G(d) level, Pople and Cur-

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E De Proft, P Geerlings/Chemical Physics Letters 262 (1996) 782-788

tiss [5] found SiH~- to be a donor-acceptor complex of SiH~- with H2 showing Cs symmetry; the bond length for H2 was found to be 0.794/~,. However, the Car structure was found to be a local minimum 7.1 kcal/mol above the global minimum. It thus seems that the potential energy surface of the silane radical cation is heavily dependent on the calculation method and basis set used. This makes it a good test case for the performance of density functional methods, for which much interest has existed recently. Moreover, the system consitutes a beautiful example of the difference in chemical properties of first and second row compounds (i.e. CH4 versus Sill4), for which we had and continue to have a lot of interest (for a review, see Ref. [ 13] ). In this contribution, the Jahn-Teller distortion of the silane radical cation is studied using ab initio and density functional methods in combination with Dunning's correlation consistent cc-pVTZ basis set [ 14]. In the discussion, use is made of an energy component analysis proposed by Boyd et al. [6].

2. C o m p u t a t i o n a l

methods

All calculations were performed using the Gaussian 94 program [ 15], running on the Cray J916/8-1024 of the Brussels Free Universities computer centre. The C2v, Car and D2d and Cs (i.e. the donor-acceptor complex) structures of SiH~', together with the neutral Sill4 were calculated at both ab initio and density functional levels. At the ab initio level, we considered the Hartree-Fock method, second order M011er-Plesset perturbation theory (MP2) [ 16] and quadratic configuration interaction with all single and double substitutions [ 17] (QCISD, actually an approximation to coupled cluster theory with all single and double substitutions (CCSD)). At the equilibrium QCISD geometries, CCSD(T) [18] (CCSD with a quasiperturbative inclusion of triple excitations) calculations were performed. For these correlated methods, the frozen core approximation was used throughout. Among the DFF methods, the following were considered: - The local density approximation (LDA), using Slater's functional for exchange [19] (S) and Vosko, Wilk and Nusair's functional for correlation (VWN) [20] ; LDA thus actually implies S-VWN.

783

The gradient corrected functionals B-P86 and BLYE both using Becke's 1988 gradient corrected exchange functional [21] and Perdew's 1986 [22] or Lee, Yang and Parr's propositions for the correlation functional [23] - T h e "exact exchange" functionals B3LYP and B3PW91 [24], which are actually semi-empirical functionals proposed by Becke [25] based on the adiabatic connection theorem [26]. For all calculations, Dunning's cc-pVTZ (correlation consistent polarized valence triple zeta, a [4s3p2dlf/3s2pld] contraction of a (10s5p2dlf/ 5s2pld) primitive set) basis set [ 14] was used. At all levels, an energy component analysis was performed. The molecular energy E can be expressed -

as

E = (r) + (Vee) "['- (VNe) "4- VNN,

(1)

with (T) the average kinetic energy, (Vee) the average electron-electron energy, which, within the KohnSham formalism [27], can be written as

1 f p(r)p(r') ~7-~ +

(v~) = ~

+ ( T - T~),

exc[p(,-)] (2)

consisting of a classical Coulombic repulsion of two charge distributions p(r) and p(r'), a non-classical exchange-correlation part E x c [ p ( r ) ] and finally the portion of kinetic energy (T - Ts) not accounted for in the Ts functional (Ts is the kinetic energy of a system of non-interacting electrons). (Vr%) is the average nuclear-electron attraction energy: (VNe) =/

p(r)v(r) dr,

(3)

and VNNthe nuclear repulsion energy. As can be seen, upon comparison with high-level correlated calculations, these quantities form a good test for the ground state electron density p(r), whereas (Vee) is also a good test for the performance of exchange correlation functionals. Furthermore, frequency calculations were performed for the lowest energy geometry at all levels in order to compare the accuracy of the different exchange correlation functionals with ab initio correlated methods.

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E De Proft, P. Geerlings/Chemical Physics Letters 262 (1996) 782-788

3. Results and discussions

The ground state electronic configuration of silane is ( lal )2(2al )2( lt2)6(3al)2(2t2)6. The removal of one t2 electron from the tetrahedral molecule produces five electrons in the triply degenerate HOMO, resulting in a Jahn-Teller distortion from tetrahedral symmetry. The geometries of the resulting structures C2v, C3v D2d and Cs (the planar SiH~--H2 donor-acceptor complex), depicted in Fig. 1, were fully optimized at the respective ab initio and DFT levels. The values of the geometrical parameters indicated on this figure can be found in Table 1. The performance of DFT methods in the calculation of geometries of charged species has not been much studied, so a detailed analysis of these results is certainly warranted. The C3v structure consists of a nearly planar Sill3 fragment and an almost dissociated H atom. The geometry with the lowest energy is, in all cases, the Cs structure, consisting of a complex between SiH~- and H2. At the bottom of the table, the mean absolute deviation from the QCISD/cc-pVTZ geometry is given for all methods for both bond distances and angles. As could be expected, the Hartree-Fock method produces the worst geometries, followed closely by the local density approximation (LDA). For the bond distances, the best

D2d

C3v

cQ R

_

C2v

R 3 ( ~ c~1

Cs

Fig. I. Geometries of the silane radical cation with indication of the geometrical parameters.

values are produced by the B3LYP method, followed closely by MP2 and B3PW91. Intermediate accuracy is provided by the gradient corrected methods BP86 and BLYP. For the bond angles however, the same conclusions do not hold. In this case, MP2 gives the lowest deviations from the high level correlated calculations, followed by the gradient corrected and exact exchange density functionals. The overall worst performance is again produced by the Hartree-Fock method, followed again by LDA, although a larger gap exists between the two methods than for the bond distances. Based on these findings, it can be concluded that the density functional methods beyond the local density approximation are capable of producing reliable geometries for cations. In a second step, the ionization energy for silane was calculated using the different geometries for the silane cation. No thermal corrections were applied. The resuits for the different methods are depicted in Table 2. CCSD(T)/cc-pVTZ calculations were performed at the QCISD/cc-pVTZ geometry to serve as a point of reference for the results at the other levels. The highest energy structure for the cation is the Ded geometry, followed by the C3v and C2v structures. The lowest energy geometry corresponds to the complex between SiH~- and H2. At the Hartree-Fock level, the energy sequence between the Car and C2v structures are switched, while at the MP2 level, the correct sequence is generated. For the LDA and the gradient corrected methods, the same inversion occurs for the D2d and C3~ structures. At the LDA and BP86 levels, the sequence is inverted, although the gap becomes smaller when applying a gradient correction. For BLYP, both structures have the same energy. Again, the exact exchange methods B3LYP and B3PW91 show the best agreement with the high level correlated results. The fact that the gradient corrected functionals perform less well could be due to the fact that the gradient correction might be less well suited to describe cations, since these species can be expected to show high gradients of the electron density. The same conclusion was already drawn by the present authors for the case of anions [28]. Agreement with the CCSD(T) results is particulary good for B3LYP and for the C3~, C2v and Cs structures. An energy decomposition analysis was performed for the different levels with the cc-pVTZ basis set. The results for the QCISD level are shown in Table

E De Proft, P Geerlings/Chemical Physics Letters 262 (1996) 782-788

785

Table 1 Geometric parameters of the different structures of the silane radical cation, calculated at the ab initio and DFr levels with the cc-pVTZ basis set. Bond lengths are in Angstrom, angles in degrees. In the bottom lines, the mean absolute deviation from the QCISD values is given for the bond lengths and angles respectively. Structure

Parameter

HF

MP2

QCISD

LDA

BP86

BLYP

B3LYP

B3PW91

D2d

R

C3v

RI R2

C2v

Ri R2 ~t ~2 R

1.517 97.6 2.052 1.459 93.1 1.455 1.575 125.1 54.9 1.517 97.6 1.465 1.980 1.945 120.4 22.4 89.1

1.515 97.5 1.959 1.463 93.1 1.454 1.631 142.1 32.2 1.515 97.5 1.468 1.895 1.848 121.3 24.0 89.2

1.523 97.8 1.930 1.468 93.1 1.459 1.642 142.7 31.8 1.523 97.8 1.474 1.905 1.855 121.4 24.0 89.2

1.532 98.0 1.826 1.491 92.9 1.476 1.633 141.4 35.2 1.532 98.0 1.491 1.829 1.757 123.5 26.9 89.7

1.532 98.3 1.876 1.488 93.1 1.474 1.651 142.1 33.1 1.532 98.3 1.492 1.909 1.831 121.9 24.7 89.3

1.529 98.4 1.908 1.483 93.3 1.469 1.654 142.0 32.7 1.529 98.4 1.490 1.969 1.878 121.1 23.5 88.9

1.521 98.2 1.908 1.473 93.3 1.461 1.638 141.4 32.9 1.521 98.2 1.479 1.935 1.860 121.2 23.8 89.1

1.523 98.1 1.899 1.477 93.2 1.465 1.638 141.6 33.1 1.523 98,1 1.481 1.894 1.828 121.7 24.6 89.4

0.382 43.6

0.082 1.4

0.000 0.0

0.355 10.7

0.151 3.6

0.167 3.4

0.075 3.5

0.095 3.7

D2d Cs

RI R2 R3 st ~2 ~3

Table 2 Ionization energy (eV) for silane calculated using the cc-pVTZ basis set at the different geometries of the silane radical cation

D2d C3v C2v Cs

HF

MP2

CCSD(T)/ QCISD

LDA

BP86

BLYP

B3LYP

B3PW91

11.67 10.57 10.74 10.10

12.20 11.53 11.28 10.87

12.10 11.68 11.35 10.96

11.98 12.08 11.54 11.36

11.60 11.63 11.15 10.87

11.43 11.43 11.07 10.71

11.82 11.67 11.33 10.96

11.78 11.62 11.22 10.90

Table 3 Energy decomposition analysis of the different geometries of the silane radical cation conducted at the QCISD level with the cc-pVTZ basis set. All values are in au

D2~ C3v C2v Cs

(r)

(v~)

(vNc)

(v~)

e

290.992773 291.006037 291.022144 291.026815

127.315386 126.693434 127.178790 126.091043

-730.053779 -729.091507 -730.017052 -727.820925

20.757386 20.386555 20.799484 19.671504

-290.988234 -291.005480 -291.016635 -291.031563

3. A n a n a l o g o u s s t u d y was p e r f o r m e d by B o y d et al.

w h i c h is c o n s i s t e n t w i t h a c o n t r a c t i o n o f the e l e c t r o n

o n the m e t h a n e radical cation. It was f o u n d there that

cloud. A s can be seen f r o m Table 3, the s a m e is not

the J a h n - T e l l e r e f f e c t leads to an i n c r e a s e in (Vr~e) (i.e. a m o r e n e g a t i v e value) and an increase in (V~),

true for the silane radial cation. In this case, a decrease in (VNe) is e n c o u n t e r e d u p o n the d i s t o r t i o n o f

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E De Proft, P. Geerlings/Chemical Physics Letters 262 (1996) 782-788

Table 4 Energy decomposition analysis of the different geometries of the silane radical cation conducted at the B3LYP level with the ce-pVTZ basis set. All values are in au

D2d C3v C2v Cs

(T)

(V~)

(~)

(Vss)

E

290.513269 290.515430 290.533528 290.532296

126.934829 126.488869 126.804997 125.615865

--729.712711 -728.728197 -729.633399 -727.216224

20.780661 20.234425 20.792984 19.552611

--291.483952 -291.489473 -291.501890 -291.515452

Table 5 Energy decompositionanalysis of the different geometries of the silane radical cation conducted at the B3PW91 level with the cc-pVTZ basis set. All values are in au

D2d C3v C2v Cs

290.545519 290.548142 290.567500 290.569352

126.969015 126.604487 t 26.908909 125.879448

--729.693141 -728.805718 -729.692460 -727.608483

20.747179 20.215779 20.763973 19.695925

--291.43143 -291.43731 -291.45208 -291.46376

Table 6 Vibrational frequencies (cm -1) of the Cs structure of the silane radical cation at the different levels with the cc-pVTZ basis set. In the bottom line, the mean absolute deviation from the QCISD frequenciesis listed for each method

AII AI A" AI A~ A~ A~ A" AI

HF

MP2

QCISD

LDA

BP86

BLYP

B3LYP

B3PW91

164 587 656 752 947 1066 2359 2422 4140

237 656 691 809 891 1145 2315 2383 3895

246 640 681 795 873 1116 2263 2331 3822

307 577 709 788 928 1233 2157 2234 3227

268 607 667 758 841 1076 2151 2230 3552

241 592 638 690 846 969 2136 2218 3707

247 614 662 738 867 1040 2209 2286 3779

262 630 681 789 863 1123 2222 2296 3675

92

30

0

126

73

81

36

30

the silane radical cation towards the m i n i m u m energy structure. At the same time, the average kinetic energy of the electrons increases and the nuclear repulsion is the smallest. Moreover, in this case, a decrease in the electron-electron repulsion energy is noticed; the Jahn-Teller distortion of the silane radical cation thus gives rise to an extension of the electron cloud. Moreover, the conclusion from Boyd's work that the electron-nuclear attraction energy is the dominant factor does not hold also for this molecule. In the next tables, the values for the different quantities are listed for the B3LYP and B3PW91 methods, since these are

the only methods giving the correct energy separations between the different geometries of the cation (Table 2). The same conclusions hold for the QCISD resuits; however, the only quantity showing almost complete parallellism with the energy separations is, as in the case of the QCISD calculations, the kinetic energy. The Jahn-Teller distortion of the species tends to maximize the kinetic energy of the electrons in the ion, while decreasing the electron-nuclear, nuclearnuclear and electron-electron repulsion energy. Finally, the IR properties of the radical cations will be investigated. At all DFT levels (vide infra), the

E De Proft, P Geerlings/Chemical Physics Letters 262 (1996) 782-788

D2d and C3v geometries have two imaginary frequencies. The C2v structure has one imaginary frequency; this structure thus corresponds with a first order saddle point. As said, the lowest energy geometry can be looked upon as a complex between SiH~- and H2. This is proven by the population analysis of this molecule performed at the QCISD/cc-pVTZ level using Cioslowski's atomic polar tensor (APT) method. The charges on the different atoms are +0.9904 on Si, -0.0438 on the both hydrogen atoms forming the Sill2 part and +0.0363 and +0.0609 on the other two hydrogen atoms. The positive charge is almost entirely located on the silicon atom. It is interesting to calculate the energy change for the reaction SiH~-(2A ') --* SiH~-(2A1 ) + H2(l~g).

(4)

The reaction energies at zero kelvin (thus including zero-point vibrational energy correction) for this reaction at the different levels are 22.87 kcal/mol (LDA), 13.62 kcal/mol (BP86), 10.21 kcal/moi (BLYP), 10.64 kcal/mol (B3LYP) and 12.70 kcal/mol (B3PW91). Using G2 theory [29], which actually corresponds to a QCISD(T)/6311 + + (2df, p) / / M P 2 ( F U ) / 6 - 3 1 G ( d ) calculation, a reaction energy of 7.63 kcal/mol is obtained. It also has to be realized that all these reaction energies are overestimated since no correction was introduced for the basis set superposition error: The complex SiH~-H2 will thus be over-stabilized. This means that the dissociation of SiH~- in SiH~- and H2 only takes a few kcal/mol. This is probably the reason why no experimental data are available yet for the ion; it can easily dissociate in SiH~- and H2. Finally, the vibrational frequencies of the silane radical cation are listed in Table 6 at all levels. The largest mean absolute deviation from QCISD for the nine modes occurs for the LDA method followed by Hartree-Fock. The gradient corrected methods perform somewhat better than Hartree-Fock, but the overall best performance is due to MP2, B3LYP and B3PW91. B3PW91 would be the best methods if it did not put in such a bad performance for the mode with the highest frequency. This frequency, which corresponds to the streching of the H2 molecule in the SiH~--H2 complex seems to be very sensitive to the introduction of electron correlation. Leaving this mode out of the statistical treatment of the result, B3PW91

787

certainly shows the best overall performance in the calculation of the vibrational spectrum of this ion.

4. Conclusions The Jahn-Teller distortion of the silane radical cation was studied at the ab initio and density functional levels using Dunning's cc-pVTZ basis set. This cation has a threefold degenerate HOMO containing five electrons, resulting in distortion possibilities towards geometries having D2~, C3v, C2v and Cs symmetry. These structures were fully optimized at both ab initio and density functional levels and were investigated using an energy component analysis. It was found that upon Jahn-Teller distortion towards the strucure with the lowest energy, the kinetic energy increases and the electron-nuclear attraction energy, electron-electron repulsion energy and nuclearnuclear repulsion decrease. The exact exchange density functional methods B3LYP and B3PW91 were found to be the methods best suited for the study of geometries, energies and vibrational spectra of these cations.

Acknowledgement FDP wishes to acknowledge the Belgian National Fund for Scientific Research (NFWO) for a post doctoral fellowship.

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