Renner–Teller vibronic potential constants for HCO

Journal of Molecular Structure (Theochem) 423 (1998) 245–250

Renner–Teller vibronic potential constants for HCO Martin Breza*, Martina Bittererova´ Department of Physical Chemistry, Slovak Technical University, Radlinske´ ho 9, SK-812 37 Bratislava, Slovak Republic Received 10 January 1997; accepted 2 April 1997

Abstract The adiabatic potential surface of HCO obtained by the single-reference ACPF method is quantitatively described using re-derived formula based on the treatment used for Jahn–Teller systems. The harmonic force constant related to the bending vibration is negative with a lower absolute value than the corresponding vibronic constant. An insufficient number of potential constants related to bending vibrations may lead to qualitatively incorrect results. q 1998 Elsevier Science B.V. Keywords: Renner–Teller effect; Perturbation theory; Evaluation of potential constants

1. Introduction Vibronic interactions in orbitally degenerate electronic states of linear triatomic molecules were first studied by Renner [1]. The effects resulting from the presence of orbital angular momentum are usually referred to as the Renner–Teller effect. As a consequence of the vibronic interaction, the linear configuration of nuclei in the degenerate electronic state is unstable. Renner’s original formulae were obtained on the basis of two harmonic potential curves which become degenerate at the linear configuration [1,2]. Recently, a number of accurate experiments have revealed small discrepancies which indicate that Renner’s original theory is not quite complete. Some discrepancies may be removed by introducing a small correction factor [3,4] and/or anharmonic potential terms into the theory [5]. Further improvement of the theory

* Corresponding author. E-mail: [email protected]

must lie in a refinement of the assumed shapes of the adiabatic potential surface (APS) and/or in a more accurate representation of the bending motion itself [2]. The usual treatment of APS evaluation may be summarized as follows [6]. Linear ABC molecules belong to the continuous C `v group. Their maximum electronic degeneracy is double. The electronic factor in the wave function can be written for degenerate states as F(r) = f (r)exp{imv}

(1)

in terms of the angle v which represents the angular whereabouts of the electron with respect to an arbitrary plane containing the linear molecule. For m = 6 1, 6 2, 6 3, … the degenerate E m state is usually designated as p, d, f, etc. [6]. Vibrations which bend the molecule out of the linear position are also doubly degenerate. There may be a number of such modes. However, in a triatomic molecule there is just one such mode to which the angular frequency is conventionally assigned

0166-1280/98/$19.00 Copyright q 1998 Elsevier Science B.V. All rights reserved PII S 0 16 6- 1 28 0 (9 7 )0 0 14 8 -6

246

M. Breza, M. Bittererova´/Journal of Molecular Structure (Theochem) 423 (1998) 245–250

and which can be given by the polar representation as the orthogonal pair of modes (qcosf, qsinf). The bending angular parameter q is expressed in units of the zero point motion amplitude. The angle f is that subtended between the plane of the instantaneously bent molecule and the arbitrary plane introduced earlier. The first task now is to derive the potential surfaces for the bending vibration when the electron is in the position (r, v). By physical considerations, the potential will depend on the difference b = v − f and will be an even function of this argument. The potential may be developed as a Fourier series V (q, r) = ∑ an (r)qn cosnb

(2)

n

If we calculate the expectation values V + and V − of Eq. (2) within the even ( + ) and odd ( − ) states, only even powers of q survive the integration over v. For the system in an E m state, V + or V − start with q 2m. For the E 1 electronic state, the simple formula 

V=



(3)

2. Analytic shape of APS In perturbation theory treatment, the exact Hamiltonian may be divided into an unperturbed part H 0 and the perturbation H9. In our case, H 0 corresponds to the system without any electron degeneracy and its APS may be described by the Taylor expansion around the reference system in optimal C `v geometry and V 0 potential value 1 1 1 9 2 2 V = V0 + K19 S12 + K29 S22 + K12 S1 S2 + K39 (S3x + S3y ) 2 2 2

+

2

∑

i#j#k = 1

9 9 9 2 2 Tijk Si Sj Sk + (T133 S1 + T233 S2 )(S3x + S3y )

2

∑

i#j#k#l = 1

Q9ijkl Si Sj Sk Sl +

2 2 2 + S3y ) +… + Q93333 (S3x

S1 = R 1 − R 1, opt S2 = R 2 − R 2, opt S3x = (q − p)cosb = p cosb

(5)

S3y = (q − p)sinb = p sinb where K9, T9, Q9 are harmonic, cubic and quartic force constants; R 1 and R 2 are A–B and B–C interatomic distances; q = p + p is the A-B-C bond angle; R i,opt correspond to optimal linear geometry. S 1 and S 2 coordinates are of S + symmetry, whereas S 3x and S 3y coordinates correspond to E 1 symmetry. The combinations of Si Sj , Si Sj Sk , Si Sj Sk Sl , etc. fulfill the fullsymmetry S + condition. The occurrence of electron degeneracy (E 1 term) brings the correction l to the APS that may be evaluated by degenerate perturbation theory treatment det{Vkl9 − ldkl } = 0

(6)

with

1 K 6 f q2 + kq4 2

is generally accepted [7,8]. The aim of this study is a quantitative APS description for the HCO molecule using re-derived formula based on the treatment used in previous papers on Jahn–Teller systems [9–11].

+

as ]V =]Si = 0 (the condition of optimal geometry) and

2

2 2 ∑ Q9ij33 Si Sj (S3x + S3y )

i#j = 1

… 4†

V 9 = ∑ Ci Xi

(7)

i

where X i are reduced matrix elements and C i are matrices of normalized coupling coefficients. Because of renormalization, the same results are obtained using the Racah V-coefficients, Clebsch–Gordan coefficients or Wigner 3j-symbols [12–15]. For the double degenerate E 1 state of the C `v symmetry group (k, l = E1x , E1y ) the non-zero C i matrices may be of types C1 =

1

0

0

1

!

… S+

1 0 C2 =  2 1

p

1

p

0

1 1 C3 =  2 0

!

… e2 « < xy

0

1

(8)

!

… e2 v < x2 − y2

in accordance with the group-theoretical condition of the symmetrized direct product for non-zero matrix elements in the space of E m electron functions [Em × Em ] = S + + E2m

(9)

Consequently, only the vibrations of E 2m and

247

M. Breza, M. Bittererova´/Journal of Molecular Structure (Theochem) 423 (1998) 245–250

(full-symmetric) S + symmetries may produce nonzero H kl 9 matrix elements (]H/]S i and S i operators being of the same symmetry for full-symmetric H). Eq. (6), Eq. (7) and Eq. (8) imply the secular equation

and after trivial substitutions for S 3x, S 3y and APS parameters, we obtain

det{C1 X (S + ) + C1 X (e2 «)2 + C3 X (e2 v) − lC1 } = 0

V = V0 6 p2 lA2 + A4 p2 + B1 S1 + B2 S2 + B11 S12 + B22 S22

reference one Xi = ]V 9=]Si = 0 for i = 1, 2:

(10)

1 1 + B12 S1 S2 + …l + K1 S12 + K2 S22 + K12 S1 S2 2 2

with eigenvalues +

l1, 2 = X (S ) 6

r 1 1ÿ 2 2

2

X (e2 «) + X (e2 v)

9 2 9 2 9 X (S + ) = Y19 S1 + Y29 S2 + Y11 S1 + Y22 S2 + Y12 S1 S2 9 2 2 + Y33 (S3x + S3y )+

2

∑

i#j#k = 1

+

2

∑

i#j#k#l = 1

9 Yijk Si Sj Sk

9 Yijkl Si Sj S k Sl

2

9 2 2 2 ∑ Yij33 Si Sj (S3x + S3z )

i#j = 1

9 2 2 2 + Y3333 (S3x + S3z ) + …

X (e2 «) = 2S3x S3y Z = Z p2 sin 2b 2 2 − S3y )Z = Zp2 cos 2b X (e2 «) = (S3x

(12)

where 2 2 Z = A92 + A94 (S3x + S3y ) + B91 S1 + B92 S2 + B911 S12 + B922 S22

+ B912 S1 S2 + …

+

2

i#j#k = 1

2

∑

∑

i#j#k#l = 1

Tijk Si Sj Sk + (T133 S1 + T233 S2 )p2

Qijkl Si Sj Sk Sl +

2

∑ Qij33 Si Sj p2

i#j = 1

…15†

+ Q3333 p + … 4

Its simplified form corresponds to Eq. (3).

3. Methods

9 9 2 2 + (Y133 S1 + Y233 S2 )(S3x + S3y )

+

1 + K 3 p2 + 2

(11)

The reduced matrix elements X i may be evaluated in the form of a Taylor expansion analogously as in the non-degenerate case, but the resulting polynomials must be of the same symmetry as the corresponding C i matrices (alternatively, 3nj-coefficients may be used for the construction of new matrices [14]).

(14)

…13†

is a polynomial of S + symmetry (compare Eq. (4)) and Y9, A9, B9 are simple multiples of corresponding V9 derivatives that are usually denoted as vibronic parameters (or constants) of APS [9–11]. If the optimal linear geometry is chosen as the

Single-reference ACPF [16] calculations for the HCO radical in the ground and first excited electronic states within C `v and C s symmetry groups were performed [17]. All possible configurations generated by single and double excitations from the valence orbitals of the reference configuration were constructed. The core orbitals 1a9 and 2a9 were kept frozen in all calculations. The computations were carried out using the columbus program package [18]. The ANO basis sets [19] used are characterized by the two contraction schemes A and B of Table 1. Calculations were carried out for a grid of geometries in the vicinity of the optimal structures within the C s ˚ and C `v symmetry groups. Variations of 6 0.025 A and 6 28 were used for bond lengths and bond angles, respectively. The energy cut-off was 10 −5 eV. The Table 1 Basis sets a Atom

Primitive set

Contraction scheme

H

A B

8s 4p 3d

3s 2p 1d 4s 3p 2d

C, O

A B

14s 9p 4d 3f

5s 4p 2d 1f 6s 5p 3d 2f

a

From Ref. [19].

248

M. Breza, M. Bittererova´/Journal of Molecular Structure (Theochem) 423 (1998) 245–250

Table 2 Ab initio ACPF computed bond lengths, bond angles and energies of the HCO structure a ˚) Bond (A

Reference geometry Basis set A Basis set B Experiment [21] [22] Minimum geometry Basis set A Basis set B Experiment [22] [23] a b

Energy b (eV)

Bond angle (8)

C—O

C—H

H-C-O

1.188 1.185 1.186 1.186

1.065 1.063 1.044 1.064

180 180 180 180

0 0

1.183 1.179 1.175 1.175

1.117 1.115 1.125 1.119

124.5 124.6 125.0 124.4

−1.1045 −1.0973 −1.078

From Ref. [17]. Related to the reference linear geometry.

Table 3 Calculated values of APS constants for basis set A (184 APS points) Parameterization a −2

A 2 [eV rad ] ˚ −1 rad −2] B 1 [eV A ˚ −1 rad −2] B 2 [eV A ˚ −2 rad −2] B 11 [eV A ˚ −2 rad −2] B 22 [eV A ˚ −2 rad −2] B 12 [eV A A 4 [eV rad −4] ˚ −2] 0.5K 1 [eV A ˚ −2] 0.5K 2 [eV A −2 ˚ K 12 [eV A ] 0.5K 3 [eV rad −2] ˚ −3] T 111 [eV A ˚ −3] T 112 [eV A ˚ −3] T 122 [eV A ˚ −3] T 222 [eV A ˚ −1 rad −2] T 133 [eV A ˚ −1 rad −2] T 233 [eV A ˚ −4] Q 1111 [eV A ˚ −4] Q 1112 [eV A ˚ −4] Q 1122 [eV A ˚ −4] Q 1222 [eV A ˚ −4] Q 2222 [eV A ˚ −2 rad −2] Q 1133 [eV A ˚ −2 rad −2] Q 2233 [eV A ˚ −2 rad −2] Q 1233 [eV A Q 3333 [eV rad −4] Rw a

3/22 1.0(3) −1(4) 1(4)

48(10) 19(4) −0(9) −0.2(6) −1.4(19) × 10 2 −0.8(16) × 10 2 0.1(17) × 10 2 −0.1(6) × 10 2 −0(12) 2(14) −2(6) × 10 3 −1(5) × 10 3 −1(3) × 10 3 0(3) × 10 3 5(9) × 10 2 0(13) × 10 1 −3(10) × 10 1 −0(2) × 10 2 0.1(4) 0.033714

Number of vibronic constants/total number of APS constants.

7/17

7/26

1.21(4) −1.6(5) 0.0(10) 4(7) 7(7) 6(8) −0.11(2) 44(3) 16.0(10) −1(3) −0.07(2) −1.2(7) × 10 2 −0.4(7) × 10 2 −0.1(4) × 10 2 −89(17) −0.9(3) −0.3(5)

1.734(16) −1.21(16) 0.4(3) 0(2) 2(2) −3(2) −0.688(10) 43.4(10) 20.3(4) −1.7(8) −0.673(17) −121(12) −2(14) −9(10) −36(4) −1.1(2) −1.2(4) 2(5) × 10 2 1(3) × 10 2 2(3) × 10 2 0(16) × 10 1 0.2(6) × 10 2 −1(3) −1(3) 6(4) 0.667(11) 0.003630

0.031539

M. Breza, M. Bittererova´/Journal of Molecular Structure (Theochem) 423 (1998) 245–250

249

Table 4 Calculated values of APS constants for basis set B (186 APS points) Parameterization a −2

A 2 [eV rad ] ˚ −1 rad −2] B 1 [eV A ˚ −1 rad −2] B 2 [eV A ˚ −2 rad −2] B 11 [eV A ˚ −2 rad −2] B 22 [eV A ˚ −2 rad −2] B 12 [eV A A 4 [eV rad −4] ˚ −2] 0.5K 1 [eV A ˚ −2] 0.5K 2 [eV A ˚ −2] K 12 [eV A 0.5K 3 [eV rad −2] ˚ −3] T 111 [eV A ˚ −3] T 112 [eV A ˚ −3] T 122 [eV A ˚ −3] T 222 [eV A ˚ −1 rad −2] T 133 [eV A ˚ −1 rad −2] T 233 [eV A ˚ −4] Q 1111 [eV A ˚ −4] Q 1112 [eV A ˚ −4] Q 1122 [eV A ˚ −4] Q 1222 [eV A ˚ −4] Q 2222 [eV A ˚ −2 rad −2] Q 1133 [eV A ˚ −2 rad −2] Q 2233 [eV A ˚ −2 rad −2] Q 1233 [eV A Q 3333 [eV rad −4] Rw a

3/22 1.1(5) −1(7) 1(6)

39(12) 18(9) 1(11) −0.8(9) −1.3(10) × 10 2 0.1(19) × 10 2 −0.2(17) × 10 2 0.6(8) × 10 2 −1(19) 4(19) 2(5) × 10 3 −2(6) × 10 3 1(4) × 10 3 0(5) × 10 3 2(3) × 10 3 −0(3) × 10 2 −0.8(15) × 10 2 0(3) × 10 2 0.6(9) 0.058591

7/17

7/26

1.46(6) −1.2(9) 5.6(15) 9(14) −26(10) −2(11) −0.61(4) 46(3) 11(2) 1(4) −0.27(4) −1.1(6) × 10 2 0.1(9) × 10 2 0.3(6) × 10 2 −2.1(3) × 10 2 −0.9(5) 3.9(8)

1.72(6) −1.2(9) 0.7(14) −1(16) −0(9) −3(12) −0.68(14) 42.6(12) 19.9(8) −0.9(11) −0.65(6) −112(10) −13(19) 8(17) −43(7) −0.8(10) −1.4(14) 9(5) × 10 2 −1(6) × 10 2 0(4) × 10 2 1(5) × 10 2 1(3) × 10 2 3(16) 1(11) 1(15) 0.65(4) 0.058578

0.051943

Number of vibronic constants/total number of APS constants.

APS parameters were obtained using the weighted orthogonal distance regression method (software package odrpack) [20]. Bond lengths R 1 and R 2 correspond to C–O and C–H distances, respectively. The quality of the regression was measured by statistical characteristics like the standard deviations of individual potential constants and the weighted discrepancy R w factor (defined as the square root of the weighted residual sum of squares divided by the sum of the function value squares).

4. Results and discussion The results for the reference and minimum geometry of HCO are collected in Table 2. Comparison with experimental data indicates very small errors only (especially for basis set B). Table 3 and Table 4 show calculated values of APS constants in significant

numbers of digits (standard deviations in parentheses are related to the last digit). Despite various parameterizations of Eq. (15) producing slightly different results, important conclusions may be drawn. Primarily, Eq. (15) represents a very good approximation for Renner–Teller APS, as indicated by the statistical characteristics (the discrepancy R w factor is comparable with the one for Jahn–Teller systems [9–11]). However, the incomplete parameterization may lead to qualitatively incorrect results (e.g. the parameterization according to the first columns of Table 3 and Table 4 exhibits no bending minimum). Here, it must be mentioned that parameterizations neglecting higher order vibronic constants (A 4 and so on) suppose strictly quadratic dependence of energy difference between potential sheets (V + –V −) on the bending magnitude (compare Eq. (3)). This may be important for systems with larger deviations from linearity. Further improvement of the results

250

M. Breza, M. Bittererova´/Journal of Molecular Structure (Theochem) 423 (1998) 245–250

should demand larger basis sets as well as lower energy cut-off for the APS points calculation. The number of APS points is not so important as their proper distribution over the relevant APS positions. The values of the K 1 harmonic constants are almost twice as high as the K 2 constants. This is in agreement with the relative weights of the atoms included in corresponding C–O and H–C vibrations, respectively. The value of the K 3 harmonic constant is most probably negative with the absolute value lower than the A 2 vibronic constant. As a consequence, the values 1 (]2 V =]S3x, y )ref = K3 6 A2 2

(16)

are positive for the upper and negative for the lower APS sheet. Such a behaviour need not be explained by the quartic constants’ contribution (Q3333 , A4 ). The negative value of K 3 (contrary to Jahn–Teller systems [9–11]) may be explained by the negative Y 339 contribution of Eq. (12) caused by electron degeneracy into the positive K 39 value of Eq. (4) for the (unperturbed) non-degenerate systems. Similar relations hold for the Q 3333 and A 4 potential constants (despite large standard deviations for quartic constants). Consequently, the equal value of the quartic parameter k of Eq. (3) for both APS sheets used in Ref. [8] is incorrect. The value of the B 1 vibronic constant is most probably lower than the B 2 constant. However, further trends in potential constant values are illegible due to large errors (especially for mixed and higher order terms). It may be supposed that evaluation of higher order APS constants should be connected with numerical problems (Jacobian matrix singularity) due to the insufficient number of APS points and/or their improper distribution over APS. As these constants may be important for systems with larger deviations from linearity, the method of suitable cuts of APS should be used to reduce the number of regression parameters [10]. Finally, the leading role of vibronic constants in quantitative APS description can be concluded. Despite large geometry changes in the system under study (compare bond angles), the use of perturbation treatment for APS description is justified by very small relative energy changes. Nevertheless, this

treatment is well founded for Renner–Teller potentials, as indicated by our results.

Acknowledgements The work reported in this paper has been funded by the Slovak Grant Agency, project no. 1/1395/94.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

[16] [17] [18]

[19] [20]

[21] [22] [23]

R. Renner, Z. Phys. 92 (1934) 172. D. Gauyacq, C.H. Jungen, Mol. Phys. 41 (1980) 383. J.M. Brown, J. Mol. Spectrosc. 68 (1977) 412. J.F.M. Aarts, Mol. Phys. 35 (1978) 1785. J.T. Hougen, J.P. Jesson, J. Chem. Phys. 38 (1963) 1524. R. Englman, The Jahn–Teller Effect in Molecules and Crystals, Wiley-Interscience, London, 1972. J.A. Pople, H.C. Longuet-Higgins, Mol. Phys. 1 (1958) 372. J.T. Hougen, J.P. Jesson, J. Chem. Phys. 38 (1963) 1524. P. Pelika´n, M. Breza, R. Bocˇa, Polyhedron 4 (1985) 1543. M. Breza, P. Pelika´n, R. Bocˇa, Polyhedron 5 (1986) 1607. R. Bocˇa, M. Breza, P. Pelika´n, Struct. Bonding 71 (1989) 57. J.S. Griffith, The Irreducible Tensor Method for Molecular Symmetry Groups, Prentice-Hall, Englewood Cliffs, NJ, 1962. J.S. Griffith, The Theory of Transition Metal Ions, Cambridge University Press, Cambridge, 1964. A.P. Jucys, A.A. Bandzaitis, Theory of Angular Momentum in Quantum Mechanics (in Russian), Mokslas, Vilnius, 1977. K.H. Hellwege (Ed.), Numerical Tables for Angular Correlation Computations (Landolt–Bornstein, New Series, Vol. I3), Springer, Berlin, 1968. R.J. Gdanitz, R. Ahlrichs, Chem. Phys. Lett. 143 (1988) 413. M. Bittererova´, H. Lischka, S. Biskupicˇ, Int. J. Quantum Chem. 55 (1995) 261. R. Sheppard, I. Shavitt, R.M. Pitzer, D.C. Comeau, M. Pepper, H. Lischka, P.G. Szalay, R. Ahlrichs, F.B. Brown, J.G. Zhao, Int. J. Quant. Chem. S22 (1988) 149. P.O. Widmark, P.A. Malmquist, B.O. Ross, Theor. Chim. Acta 77 (1990) 291. P.T. Boggs, R.H. Byrd, J.E. Rogers, R.B. Schnabel, odrpack Version 2.01: Software for Weighted Orthogonal Distance Regression, National Institute of Standards and Technology, Gaithersburg, 1992. J.W.C. Johns, S.H. Priddle, D.A. Ramsay, Discuss Faraday Soc. 35 (1963) 90. J.M. Brown, D.A. Ramsay, Can. J. Phys. 53 (1975) 2232. E. Hirota, J. Mol. Struct. 146 (1986) 237.