HNC, exact vibrational energies, and comparison for experiment

Volume 198, number 6

CHEMICAL PHYSICS LETTERS

23 October I992

A global ab initio potential for HCN/HNC, exact vibrational energies, and comparison to experiment Joseph A. Bentley, Joel M. Bowman, Bela Gazdy Department

ofChemistry, Emory University, Atlanta, GA 30322, USA

Timothy J. Lee NASA Ames Research Center. MS230-3, Moffett Field, CA 94035-1000. USA

and Christopher E. Dateo ’ ELORET Institute, Palo Alto, CA 94303, USA

Received 6 August 1992

An ab initio, i.e. from first principles, calculation of vibrational energies of HCN and HNC is reported. The vibrational calculations were done with a new potential derived from a tit to 1124 ab initio electronic energies, which were calculated using the highly accurate CCSD(T) coupled-cluster method in conjunction with a large atomic naturalorbital basis set. The properties of this potential are presented, and the vibrational calculations are compared to experiment for 54 vibrational transitions, 39 of which are for zero total angular momentum, J=O, and 15 of whtch are for d= I. The level of agreement with experiment is unprecedented for a triatomic with two non-hydrogen atoms, and demonstrates the capability ofthe latest computational methods to give reliable predictions on a strongly bound triatomic molecule at very high levels of vibrational excitation.

1. Introduction

The prediction, from first principles, of the structure, energetics, and dynamics of molecules is a major activity and goal of theoretical/computational chemistry. Of chemically bound molecules, hydrogen cyanide has been among the most intensively studied, both theoretically and experimentally. HCN has been a fascinating and fundamental system since the discovery, in 1963, of the stable, high energy isomer HNC. The co-existence of this isomer has made the HCN*HNC isomerization a paradigm of Hatom migration, which is ubiquitous in chemistry. Such motion, in the absence of tunneling, requires a

Correspondence to: J.M. Bowman, Department ofchemistry, EmoryUniversity, Atlanta, GA 30322,USA. ’ Mailing address: NASA Ames Research Center, MS230-3,

Moffett Field, CA 94035-1000, USA.

substantial amount of energy to surmount the barrier to isomerization. Thus, a study of the isomerization has naturally led to interest about the nature of this motion, i.e. is it regular, chaotic, or both? Several classical trajecory studies [ 1,2] have examined this issue using model potentials, and have seen evidence of both chaotic and regular motion. A variety of model quantum calculations of the vibrational motion have also been reported [ 2-41; these have either made approximations to the vibrational motion, or to the potential. There have been limited ab initio electronic structure calculations of the HCN and HNC minima and the isomerization barrier [ 5,6 1, however, no ab initio global potential has been reported prior to the present calculation. The most recent ab initio calculation [6] is the forerunner to the potential surface we describe below. The most widely used potential is a semi-empirical one due to Murrell, Carter and Halonen (MCH) [ 7 1. This po-

0009-2614/92/$ 05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved.

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tential has been used in a number of recent “exact“ vibrational calculations [8-I 3 1, however, it does have several serious deficiencies, as pointed out elsewhere [ 6,11,14,15 1. An empirically modified MCH potential has recently been reported by two of us [ 111. The modifications were able to greatly improve agreement between exact vibrational calculations and recent experiments [ 14-l 71, and also to incorporate the most recent and most accurate ab initio calculations of the saddle point geometry and energy 161. Interest in HCN has been heightened due to recent experiments on highly excited vibrational states [ 14171. One, based on stimulated emission pumping (SEP) has been reported by Wodtke and co-workers [ 14,15 1, and the other, based on Fourier transform infrared (FT-IR) experiments has been reported by IUemperer, Lehmann, Smith and co-workers [ 16,171. From the SEP experiments, the energies of highly excited bending states have been determined, while the FT-IR experiments have determined the energies of many overtone and combination stretch excited states, with little or no bend excitation. These experiments, together with earlier ones on low-lying vibrational states of HNC [ 18-201, provide a large data base that challenges the accuracy and methodology of t(leory, and in turn presents theory with an opportunity to offer understanding of the experimental results, and to make predictions of unmeasured data. In this Letter, we focus on the accuracy of the latest ab initio theoretical methods for calculating vibrational energies. Specifically, we report the calculation and fitting of a large number of ab initio electronic energies using a coupled-cluster method for the HCN/HNC system. The fitted global potential is used in exact vibrational calculations for states of total angular momentum .I=0 and .I= 1. Fifty-four calculated transition energies (up to 18532 cm-’ ) are compared with available experimental data on HCN and HNC.

2. Electronic structure calculations The singles and doubles coupled-cluster (CCSD) method [ 2 1 ] that includes a perturbational estimate of the effects of connected triple excitations, 564

23 October I992

CHEMICAL PHYSICS LETTERS

CCSD(T)

[22], was used to evaluate the electronic

energy. The CCSD(T) method has been shown to predict reliably the equilibrium structure and vibrational frequencies of several molecules [ 6,231, and to reproduce CCSDT (singles, doubles and triples coupled-cluster theory) results with high accuracy [ 241. An atomic natural orbital (ANO) one-particle basis set [25] was used in conjunction with the CCSD(T) approach since this combination has proved to be especially successful in accurately predicting the equilibrium structure and harmonic frequencies of HCN, HNC [ 61 (and their relative energetics), 0, and Nz [23]. For O1 the equilibrium bond lengths and bond angle are within f0.003 A and & 0.3”, respectively, of experiment, and the calculated harmonic frequencies are within ? 30 cm-’ of experiment. Similar results were also found for HCN and HNC. Additionally, the CCSD(T) /ANO basis set approach predicts the bond distance and fundamental frequency of triply-bonded N, to within f0.002 8, and + 1.5 cm-‘. A 4s3p2dl f AN0 contraction was used for C and N and a 4s2pld contraction was used for H [25]. This basis set will be denoted [4s3p2dlf/4s2pld]. The AN0 contracted basis set is based on van Duijneveldt’s [26] 13sSp/8s primitive sets for the heavy atoms and hydrogen, respectively. The carbon and nitrogen sp primitive basis sets were augmented with a 6d4f even-tempered sequence of polarization functions, while the hydrogen primitive basis was augmented with a 6p4d even-tempered sequence. The polarization orbital exponents are given by o!=2.5Q0 for n=O, .... k, where a0=0.07 and 0.22 for the C d and f functions; a,=O.lO and 0.30 for the N d and f functions; and (Y~=O.10 and 0.26 for the H p and d functions. Only the pure spherical harmonic components of the d and f type basis functions were included in the basis set. The C and N 1s like molecular orbitals were not included in the correlation procedure. The CCSD (T) calculations were performed on the NASA Ames NAS facility CRAY Y-MP/8256 using the TITAN #’ set of coupled-cluster programs inter#’ TITAN is a set of etectronic structure programs, written by T.J. Lee, A.P. Rendell and J.E. Rice.

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faced to the system.

MOLECULE-SWEDEN

CHEMICAL PHYSICS LETTERS

c2 program

3. Fitting the surface The surface was fit in terms of Jacobi coordinates for triatomics, r, R, y, where r is the CN internuclear distance, R is the distance between the H and the center of mass of CN, and y is the angle between the two vectors Y and R, such that y=O” and y= 180” correspond to linear HCN and linear HNC, respectively. These coordinates were chosen since they were used in the vibrational calculations, which are described below. A grid of 1334 geometries spanning the ranges O”
is an electronic structure program system written by J. AlmMf, C.W. Bauschlicher Jr., M.R.A. Blomberg, D.P. Chong, A. Heiberg, S.R. Langhoff, P.-A. Malmqvist, A.P. Rendell, B.O. Roos, P.E.M. Siegbahn and P.R. Taylor.

23 October 1992

ues of R and y. The Morse parameters were determined by a standard nonlinear least-squared method. The fit was restricted to electronic energies less than or equal to 40000 cm-’ above the local minimum for the given R-y cut; the final number of electronic energies used in the fit was 1124. This fit gave the four Morse parameters ovqr most of the rectangular grid, O”
Table I Stationary points, electronic energiesAE, and AE plus the exact zero point energy, AFCZPE, relative to the HCN minimum for the fitted potential surface a)

R (00) r (a~) Y(deg) rcN (~0) rCH(Q) FNH(a01 AE (cm-‘) AE+ZPE (cm-‘)

HCN

HNC

CHN (s.p.)

3. I94 2.196 0.0 2.1Y6 (2.179) b’ 2.01 I (2.012) 4.206 (4.191) 0.0 3571.7

2.891 2.221 180.0 2.220 (2.209) 4.087 (4.087) 1.867 (1.878) 5265.8 8621.7

2.185 2.258 75.80 2.258 2.224 2.642 17OL7.7 _

” The calculations were done in atomic units. 1 hartreez219474.6 cm-‘, lengths in bohrs (a,), and masses in atomic mass units, C= 12.0, N= 14.00307, and H= 1.007825. b, Values in parentheses are the experimental quantities given in ref. 161.

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R (a,) Fig. 1. Equipotential contour plot of the fitted HCN/HNC potential with r fixed at its equilibrium value forlinear HCN. Contour intervals are 2000cm-‘, and the maximum contour value is 20000 cm-‘.

23 October I992

These agree well with the CCSD(T) calculations reported earlier (with a slightly different basis ) [ 6 I, and also with experiment. In addition, the difference between HCN and HNC zero point energies, 5050.0 cm-‘, is in agreement with the most recent experimental estimate of 5 176 + 700 cm- ’ [ 61. Equipotential energy contour plots of the fitted potential as a function of R and y, for r fixed at its equilibrium value for linear HCN, and as a function of R and Y for y=O” and 180”, are given in figs. 1 and 2, respectively. As seen from fig. I, there is substantial correlation between the R and y modes. This large correlation causes great difficulties for conventional methods for calculating vibrational energies. The contour plots in fig. 2 also show substantial Rr correlation; however, as can be surmised from fig. 2, a simple rotation of the radial coordinates significantly reduces the correlation. We take advantage of this in one set of vibrational calculations, which are described below.

4. Vibrational calculations

:

y = 180”

Fig. 2. Equipotential conlour plots of the fitted HCN/HNC potential for the linear geometries, HCN (y=O”) and HNC (y= 180” ). Contour intervals are 2000 cm-‘, and the maximum contour value is 20000 cm- ’

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Most of the vibrational energy levels for J=O, were computed using the discrete variable rcpresentationdistributed Gaussian basis (DVR-DGB) method [ 81. This method is especially well suited for highly excited bending states. As applied here, the DVR consists of a “pointwise” basis set equivalent to the N Gauss-Legendre quadrature points in the variable X= cos y. Thus, a given DVR point, x,, defines a reduced dimensionality ray Hamiltonian in the two radial coordinates, R and Y, with the potential depending parametrically on x,. The DVR-DGB method uses a two-dimensional Gaussian basis set to obtain the ray eigenfunctions and eigenvalues of this Hamiltonian, which are truncated and then recoupled in the DVR procedure. Our largest calculation consisted of 65 DVR points combined with a twodimensional Gaussian basis set distributed over the potential up to 30000 cm- ‘. Only ray eigenfunctions with energies below 23500 cm-’ were retained for the final three-dimensional calculation. The final basis set consisted of 12 12 contracted functions. The first 93 vibrational states were converged to less than a wavenumber by comparison with smaller basis set calculations.

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CHEMICAL

PHYSICS LETTERS

All calculations for total angular momentum J= 1, and also for three HCN high-energy stretch overtone states for J=O (labeled (2, 0, 3), (0, 0, 5) and (0, 0, 6) in table 2) were done with the dressed state truncation/recoupling method [ 10,111. In this method, which is similar in strategy to the DVR, a series of “dressed” two-dimensional Hamiltonians are defined with respect to a basis over the dressing mode, r in this case. These Hamiltonians are diagonalized, and the eigenfunctions are truncated and recoupled to form a three-dimensional Hamiltonian matrix, which is diagonalized to obtain the eigenvalues and eigenfunctions. The J= 1 calculations were done for the ]KI = 1 even parity state, where K is the projection quantum number of the angular momentum on the body-fixed z axis, which is coincident with the vector R. (In the linear molecule limit, 1KI is usually denoted L’and is referred to as the vibrational angular momentum.) A basis of 50 associated Legendre polynomials was used for y, and a basis of 16 and 18 numerical functions was used for a set of rotated radial coordinates [29], for which the radial correlation is reduced. These numerical functions were contracted from 39 primitive harmonic oscillator functions for each radial mode. The primitive radial basis functions were contracted by diagonalizing one-dimensional zerothorder Hamiltonians. The recoupling included all twodimensional eigenfunctions with energies below 40000 cm-‘. Convergence to within one wavenumber or less was determined by running smaller bases, and also by doing J=O calculations and comparing energies for states with ten quanta of bend or less with those from the DVR-DGB calculations. The J= 0 calculations of the high stretch overtone states, mentioned above, were done basically as described for the J= 1 calculations, except that tight numerical functions (contracted from 50 Legendre polynomials) were used for the basis in y. The calculated transition energies arc compared with experiment for HCN in table 2, and for HNC in table 3. (The odd numbered bending states correspond to the .I= 1, 1KI = 1 calculation.) The level of agreement between the ab initio energies and experiment, especially for the highly excited states, is unprecedented for a triatomic system with two nonhydrogen atoms. There are some noteworthy results for the HCN highly excited bending states. The state

Table 2 Comparison experiment

23 October

of calculated HCN transition

Theory

(0 I,01 (O,L 0) (l,O,O) (0,3,0) (1% I,01 (0,4,0) (O,O, 1)

(1,&O) (0, 50) (0, I, 1) (2,0,0) (1,3,0) (OJ, 1) (2, 1,O) (l,O, 1) (2,&O) (074, 1) (I, 1, 1) (3,0,0) (0,0,2)

( 1,2, 1) (0, 132) (LO, 1) (1,032) (1%10,O) (3,4,0) (1,1,2) (3,Q 1) (4,&O) (O,O, 3) (1,232) (1,12,0) (0, 1>3) (3,6,0) (2,0,2) (2, l&O) (4,430) (I, 14,O) (5,&O) (1,0,3) (1,1,3) (0,0,4) (0, 1, 4) (2,0,3) (0,095) (&I, 5) (O,O, 6) “‘Ref.

[17].

energies (cm-’

Exp.

713.46 a’ 1411.42” 2096.68 a) 2114.94”’ 2807.06 a’ 2802.96 a’

3345.6 3536. I 3587.3 4040.0 4159.2 4262.2 4726.6 4896.4 5413.3 5600.8 6101.0 6108.4 6208.9 6582.1 6792.8 7244.4 7460.2 8628.6 8980.8 9045.6

331 1.48 a’ 3502.12 a’ 3496.68 a’ 4005.63 a) 4173.10”’ 4202.68 a’ 4684.3 I a) 4879.73 I’ 5393.70 a’ 5571.89 a’ 6036.96 a’ 6084.80 a’ 6228.59 ” 6519.61 a) 6761.33 a’ 7194.22 a’ 1455.42 a’ 8585.57 ” 8926.79 b’ 8995.22 b’

9307.8 9487.3

9251.53 a’ 9496.43 a’

10657.3

11006.7 11050.1 I1 533.7

9648.65 9627.08 9914.39 10227.51 10281.81 10350.11 10631.44” 10974.18

) with

Exp. - theor.

74 I .4 1446.8 2088.6 2170.9 2827.1 2875.2

9657.8 9718.8 9954.2 10263.0 10367.1 10431.7

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b’ a’ a’ b’ a’ b’ b’

11650.9

11015.86 b’ 11502.45 b, 11654.59 b’

I 1738.9 12384.3 12759.4 13366.9 13745.6 15696.4 16301.9 18532.3

11674.45 ‘) 12326.94 4’ 12635.89 ” 13270.75 aJ 13702.24 a) 15551.94”’ 16165.55 a’ 18377.01 ”

-27.9 - 35.4 8.1 - 56.0 -20.0 - 72.2 - 34. I - 34.0 -90.6 -34.4 13.9 -59.5 -42.2 - 16.7 - 19.6 -28.9 -64.0 -23.6 19.7 -62.5 -31.5 -50.2 -4.8 -43.1 -54.0 -50.3 -50.2 9.1 -9.2 -91.8 -39.8 -35.5 -85.3 -81.5 -25.8 -32.5 -34.2 -31.3 3.7 -64.5 - 57.4 - 123.5 -96.1 -43.4 - 144.5 _ 136.3 - 155.3

b1 Ref. [IS].

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Table 3 Comparison ofcalculated HNC transition energies (cm-‘) with experiment Theory

Exp.

Exp.- theor.

(0, 170) (0, 2,O) (1,0,0) (0, 0, 1)

465.7 913.1 2014.4 3606.4

477 a’ 932 b’ 2029.2=I 3652.7‘P’

11 18 14.8 46.3

(I,% I)-(l,O,O) (0, 1. 1)-CO,I,01 (O,L 1)-CO,2.0)

3617.5 3600.I 3591.9

3649.54d’ 3630.2=A) 3607.95”

32.0 30.I 16.0

a) Matrix data in ref. [ 181. b1Ref. [4]. ClRef. [ 191. ‘) Ref. [ 201.

(1, 12, 0) on the semi-empirical MCH potential is 190 cm- ’ below experiment [ 111, whereas the present calculation in only 35 cm-’ above experiment. The calculated higher excited bend state ( 1, 14, 0 ) is only 31 cm-’ above experiment, whereas on the MCH surface that state is 306 cm-’ below experiment [30]. These results support the earlier claim that the MCH barrier to isomerization is too low [141. Even though there is good agreement with experiment, there are some systematic trends in the differences between theory and experiment that can be discerned by examining the pure overtone transitions. The difference is positive and increasing for the CN stretching overtones, but negative and increasing in magnitude for both the CH stretch and bending overtones. Thus, there is some cancellation of errors in combination states comprised of CH and/ or bend excitations with CN excitation. A close examination of the errors in the calculated pure overtone transition energies reveals that the errors are nearly constant between adjacent states, and thus the errors are nearly a constant fraction of the transition energy. A relatively simple and small scaling correction (as described in ref. [ 1 1 ] ) to the present surface could bring theory and experiment to within “wavenumber accuracy”, i.e. to within 1 cm - ’ or less. It is also likely that additional ab initio electronic energy calculations will improve the fitted potential which should lead to improved agreement with experiment. We hope to do both in the future.

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Acknowledgement We thank the Cherry L. Emerson Center for Scientific Computation at Emory University for the use of computational resources. JMB acknowledges support from the National Science Foundation (CHE9200434) and CED was supported by a grant from NASA to ELORET Institute (NCC2-737).

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