A potential energy surface for the electronic ground state of N2O

6 June 1997

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 271 (1997) 157-162

A potential energy surface for the electronic ground state of N 20 Guosen Yan, Hui Xian, Daiqian Xie Department of Chemistry, Sichuan University, Chengdu 610064, People's Republic" of China Received 15 January 1997; in final form 7 April 1997

Abstract

A potential energy surface for the electronic ground state of N20 is optimized using a variational procedure with an exact vibrational Hamiltonian. In the optimization, the ab initio force field of Martin, Taylor and Lee is taken as a starting point, and the observed vibrational band origins up to 15000 cm- ~ reported by Campargue and co-workers are involved. The RMS error of this fitting to the 60 observed ~ state vibrational energy levels is 0.34 cm- ~. The rovibrational energy levels for the S and 11 vibrational states are calculated to test the refined potential. 1. Introduction

Nitrous oxide plays an important role in understanding stratospheric ozone chemistry. The rovibrational spectra of the N20 molecule can help people to detect its internuclear dynamic properties. Early in 1950, Herzberg and Herzberg [ 1] reported the experimental vibrational energy levels of the N20 molecule below 8000 cm -t. In 1974, Amiot and Guelachvili [2] studied the vibrational band origins up to 7800 cm J with a Fourier transform spectrometer. Recently, Campargue and co-workers[3] observed the near infrared and visible absorption spectrum of N20 between 6500 and 11000 c m - ~, by Fourier transform absorption spectroscopy, and between 11700 and 15000 cm -~, by intracavity laser absorption spectroscopy. These experimental results make it possible to refine the potential energy surface of N20 using the observed highly excited vibrational band origins. Nitrous oxide is a linear molecule at the equilibrium geometry of the electronic ground state. The earliest work on its internuclear potential was made by Suzuki [4], who determined the quartic force field of N20 using standard perturbation theory. In 1976,

Ch6din and co-workers [5] reported the first calculation of the sextic force field using a new algebraic approach for solving the vibrational eigenvalue problem for linear triatomic molecules. The problem was then reinvestigated by Lacy and Whiffen [6], and Kobayashi and Suzuki [7], by means of a direct numerical diagonalization method. In 1989, Teffo and Ch6din [8] investigated the internuclear potential up to sextic terms using an algebraic contact transfornmtion method. All the methods mentioned above used truncated Hamiltonians for the nuclear motion, and solved the corresponding ScbriSdinger equation approximately or semi-empirically. The PES thus obtained can not reproduce the observed rovibrational energy levels well when an exact rovibrational Hamiitonian is used. For example, we have calculated vibrational band origins using the DVR3D procedure of Tennyson and co-workers with the PES of Ref. [8]. The calculated results show that there are large discrepancies for almost all the observed vibrational band origins (please see Section 3 for details). In this Letter, we optimized the potential energy surface for the electronic ground state of the N20 molecule by means of a variational procedure using

0009-2614/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 I 4 ( 9 7 ) 0 0 4 3 5 - 1

158

G. Yan et al. / Chemical Physics Letters 271 (1997) 157-162

the exact vibrational Hamiltonian in bond lengthbond angle coordinates.

2. The form of the potential energy function and the optimization of parameters In a theoretical study of the rovibrational state for a triatomic molecule, the potential energy function is often written as a power series expansion of suitable coordinates. In the present work, the Morse-cosine expansion of Jensen and Kraemer [9] is used as the analytical representation of the potential function:

where the summation runs from i = 1 to i = N for linear molecules, and from i = 2 to i = N for nonlinear ones. fo(i) are expansion coefficients. Recently, Xie and Yah [10] presented a self-consistent field configuration interaction (SCF-CI) procedure for determining the PES of triatomic molecules from the observed vibrational band origins and refined the PES of H 2 0 [11] and NO 2 [12]. In this procedure, the optimized parameters in the potential energy function can be obtained by minimizing the weighted least-squares objective function defined as F = E [ w , ( E,(°bs) - E~ca'))] 2 ,

V(A rl2, A r32, p) = v0( ) + E J

"~ E Fjk(-P)Yj Vk + E Fjkm(-P)YjVkYm j<~k j~k<~rn +

E

+ ....

j<~k
(1)

where all the indices j, k, m and n assume the values 1 or 3. The quantity Y~ is defined as: = 1 - e x p [ - o t j ( rj - rje)],

(2)

where the a t are molecular constants and r~e ( j = 1 or 3) are the equilibrium values of rj ( j - - 1 or 3). The quantity p is the instantaneous value of the bond angle supplement ( p = 7 r - 0, 0 is the enclosed bond angle). The Fjk... are functions of p and defined as N f j ( p ) = E fj(i)( cOS D e - COSp) i i=1

(3)

and N

Fjk...(P) =f)(0?.. + E fj(ki?..(cOS Pe -- COSp) i' i=1

(4)

where symbol pe is the equilibrium value of p and (,)' are expansion coefficients. The function the fjk V0(-p') i's'the potential energy for the molecule bending with bond lengths fixed at their equilibrium values:

Vo(p) = E fo(/)( cOS Pe -- COSp) i, i

(6)

t/

(5)

where the summation runs over all the observed vibrational states of interest, w, is the weighting coefficient for vibrational state n, En(°bs) is the observed band origin and E~ca1) is the calculated band origin obtained using a variational procedure, such as the contracted method [13], the discrete variable representation (DVR) method [14] or an SCF-CI method [10]. The optimizer LMF [15] can be used to minimize the least-squares objective function. The derivatives of the calculated vibrational energies with respect to the parameters in the potential energy function OE,/OP (where P denotes any one of the parameters) can be calculated using the HellmannFeynman theorem:

0P

=

,oo-

0P

(7)

In this work, the SCF-CI procedure [10] in bond length-bond angle coordinates is used to calculate the vibrational energy levels and their first derivatives with respect to the parameters in the potential energy function. At first, we performed an SCF calculation on the vibrational ground state by solving the stretching one-dimensional SCF equations using the renormalized Numerov method of Johnson [16] with 2000 sectors in the region [0.7 au, 6.0 au], and solving the bending one-dimensional SCF equation variationally with 80 Legendre polynomials as basis functions. Then, we carried out CI calculations with two sets which take 1600 and 1800 lowest configurations, respectively. This test shows that the CI calculation with 1600 configurations converged all band

G. Yan et al. / Chemical Physics Letters 271 (1997) 157-162 Table 1 The optimized potential energy parameters for N20 (in cm ~) ,f~l)

f~o)

5328.062

- 121298.805

13809.536

j~l)

33289.317

38887.812

f~0)

77144.929

,li~~-~

73997.874

,f~2)

.1~2 )

- 120398.189

f~)

.f~l ~

- 55506.533

f~t°~

6581.754

f~o~

-1166.087

.f~}

30638.353

f~) fill )1

-24191.057 66543.732

•f'~l'~

706.299

f~0~

- 7380.562

f(~3~

8580.613

fl~4)

185740.630

,~_~

- 2321.962

.fl ~

- 190320.542

•1"~I I

- 7191.497

f ( IoI)12

1234.455

J'~l~ ~_

- 2099.590

f~°ll f~

85950.464

~m) J1222 f~)

- 4495.150

- 29173.474

.1'I I ~

- 12932.685

.fl~'

~,, r 2 = 1 . 1 8 5 1

4776.474 88145.667 ~,, c e 1 = 2 . 7 5 1

~-I

3. Results and discussion In the optimization of the parameters in the PES, 60 observed vibrational band origins up to 15000 cm-1 appearing in Ref. [7] and reported by Campargue and co-workers in Ref. [3] are used as the input data points. The ab initio force constants of Martin, Taylor and Lee (MTL) [17] are taken as the initial guess for the PES. The weighting coefficients in the least-squares objective function (6) are taken to be 1.0 for all the band origins. The non-linear parameters related to the geometry of the system were held fixed at the following values: rje = 1.1273 A, r2e = 1.1851 ,~, p = 0 ° [8] where r I and r 2 are the bond lengths of NN and NO respectively). The other parameters in the PES are optimized. The optimized parameter values of the PES for N 2 0 are given In Table 1. In Table 2, the values of the force constants are given together with the ab initio results of MTL [17], Allen and co-workers [18], and with other results obtained from the observed band origins [4,6-8]. It can be seen from Table 2 that the quadratic force constants, and the diagonal terms of the cubic to quartic force constants ca

4132.477

,f~)

o, r t = l . 1 2 7 3

699.504

- 138208.066

•1"I~! p=0

- -

j~3)

159

42 =

2.8042 A

origins below 15000 cm -~ to 0.01 cm -~ or better, compared with the results obtained with 1800 configurations. Thus, all subsequent CI calculations required in the optimization are performed with 1600 configurations.

Table 2 Force constants for N20 and comparison with other results (aJ; a ; r a d ) '~ A b initio [17]

A b initio [18]

Ji-,

18.126

21.914

18.251

18.236

•f;R

0.960

1.635

1.028

1.029

fRR

12.021

12.096

11.960

l 1.966

£ , ,,

0.683

,/~ r,-

-- 137.969

0.763 -- 161.1

T e f f o [8]

0.666 -- 133.6

K o b a y a s h i [7]

0.666 -- 132.433

S u z u k i [4]

This work

18.588

18.190

18.653

1. 134

1.024

0.718

11.802

12.031

11.693

L a c y [6]

0.666

0.661

-- 150.48

0.666

-- 134.665

-- 151.262 -- 2.548

fm •

-- 2.532

-- 1.069

-- 6 . 8 7 2

-- 9.842

-- 4 . 5 4

-- 1.920

JR R ~

- 0.295

- 0.928

1.498

2.451

- 5.90

- 0.473

fRkk

-- 9 8 . 0 0 2

-- 1 0 4 . 3 0 0

-- 9 8 . 8 3 0

-- 9 6 . 3 2 7

-- 9 4 . 7 4

,[i. . . .

- 1.691

- 1.722

- 1.580

- 2.567

- 2.918

- 1.681

J~, R

- 1.449

- 1.655

- 1.537

- 1.101

- 0.784

- 1.574

0.755

,/~ rr,-

811.360

931.1

691.4

674.748

52.56

745.291

949.184

J}e r r,

10.111

JR R,,

0.0797

,/RRRr

17.778

J'RRRR

587.169

-- 102.90

2.906 -- 9 4 . 7 4 6 - 5.944

5.699

46.65

65.608

-- 5 1 . 6 0

10.429

- 12.017

5.077

-- 3.485

8.771

171.96

-- 8.681

-- 12.719

- 7.691

- 20.759

9.84

- 12.894

- 27.299

590.330

623.28

614.926

571.393 30.755

12.17 614.0

634.9

,1~, ,~

1.105

0.237

1.808

12.670

-- 1.44

3.002

J~,~Rr

4.041

4.117

5.105

7.222

0.20

5.015

4.131

.[c, ,~ te R

2.394

2.491

1.491

-- 6.165

3.90

3.185

-- 18.765

j~,~,,,

1.985

2.414

1.897

2.282

2.652

1.812

8.159

a r = rNN, R

=

rNO.

160

G. Yan et al. / Chemical Physics Letters 271 (1997) 157-162

Table 3 Vibrational band origins and effective rotational constants of N20 for the £ states ( c m - 1)

(/"lb'2/)3)Vobs. (000) (0 2 0) (1 0 0) (0 0 1) (0 4 0) (1 2 0 ) (2 0 0) (0 2 1) (0 6 0) (I 0 1) (1 4 0) (2 2 0) (3 0 0) (002) (0 4 1) (1 2 1) (1 6 0 ) ( 2 0 1) (2 4 0) (3 2 0) (4 0 0) (0 2 2) (1 4 I) (3 0 1) (3 4 0) (4 2 0) (5 0 0) (0 0 3) (0 4 2) (1 2 2) (2 0 2) (3 2 1) (4 0 1) ( 4 0 I) (6 0 0) (6 0 0) (4 4 0) (0 2 3) (1 0 3) (1 4 2) (3 0 2) (4 2 1) (5 0 1) (4 2 I) (004) (1 2 3) (2 0 3) (4 0 2) (402) (6 0 1) (1 0 4 )

0.0 1168.13 1284.90 2223.76 2322.57 2461.99 2563.34 3363.98 3466.60 3480.82 3620.94 3748.25 3836.37 4417.38 4491.54 4630.16 4730.82 4767.14 4910.99 5026.30 5105.68 5529.69 5762.37 5974.84 6192.27 6295.44 6373.21 6580.83 6630.41 6768.48 6868.53 7024.07 7137.10 7214.65 7463.96 7556.11 7640.45 7665.22 7782.64 7998.56 8083.93 8276.30 8376.32 8452.61 8714.11 8877.03 8976.50 9219.03 9294.97 9606.30 9888.58

Robs.--//cal. a l"obs.--//cal. b By(cal.) Bv(obs.) 0.0 0.47 0.07 --0.36 0.22 0.01 0.21 0.18 0.30 --0.21 --0.19 --0.34 0.34 --0.35 --0.28 --0.07 --0.09 0.04 --0.34 --0.44 0.34 0.16 --0.20 --0.02 --0.18 --0.10 0.02 --0.16 --0.59 0.15 -- 0.14 0.09 --0.34 --0.04 0.20 0.52 -- 0.14 0.22 --0.01 0.14 -- 0.14 0.66 --0.19 0.02 0.14 0.59 0.02 0.04 --0.04 0.07 0.14

0.0 0.41880.41901101 --0.280.41940.4199209 --3.670.41720.4172550 --4.860.41500.4155565 --0.980.41970.4206156 --7.970.41780.4181477 --31.580.41570.4156055 --8.400.41560,4165460 -- 2.65 0.42000,4211936 --57.760.41340.4137725 -- 17.730.41810.4187840 --45.830.41610.4163280 -- 120.160.41430.4141560 --37.2 0.41120.4121002 -- 15.120,41610.4172990 22.320.41400.4147812 26.230,41190.4121385 --31.750.41820.4193047 --45.180.41640.4169020 39.860.41430.4144039 --81.890.41330.4130535 --50.550.41190.4131662 17.620.41440.4154690 17.190.41050.4106715 35.530.41460.4149085 121.460.41250.4123970 --43.230.41260.4123420 -- 11.600.40740.40863500 --64.150.41240.4139942 3.650.41040.41136473 -- I 1.280.4081 0.40862284 5.560.41280.4136395 34.770.41060.41096810 89.660.40950.40961171 118.680.41260.41279544 186.410.41080.41065351 40.530.4121 0.411845 21.480.40830.4097790 --20.380.40580.40682265 23.310.40870.409635 69.040.4067 0.40718036 -- 19.070.41080.411584 --35.010.40860.4089394 20.970.40890.4090441 --72.500.40370.40518433 -- 1.290.40670.4080025 37.91 0.4043 0.40512510 --20.300.40690.407471 --51.390.40580.406185 -- 33.690.4072 0.407236 72.810.40220.4033264

Table 3 (continued)

(/'tl/~'2/"3) l~obs, (1 4 3) 10079.56 (3 0 3) 10163.61 (5 0 2) 10429.12 (0 0 5) 10815.27 (0 4 4) 10820.14 (0 2 5) 11844.97 (1 0 5) 11964.25 (0 0 6) 12891.15 (1 0 6) 14009.69 (0 0 7) 14934.27 RMS error

l"obs.--Vcal. a bobs.--/~cal. b Bv(caL) Bv(obs.) 0.70 -0.17 - 0.74 0.35 - 0.98 -0.04 0.41 0.17 -0.25 -0.03 0.34

8.360.40520.406159 -0.560.40320.4036886 56.000.40490.4054490 57.890.40220.40424 22.320.40360.404730 31.530.40220.403781 1.580.40320.400092 0.40800.398379 0.39970.396570

Residuals calculated from the present PES. b Residuals calculated from the sextic force field of Ref. [8].

(except f ~ ) obtained in this work are almost in keeping with the values of the ab initio results of MTL [17], and the results from the observed band origins of Teffo and Ch6din [8], and Kobayashi and Suzuki [7]. All the other force constants have larger discrepancies, for example, for force constants f~ .... the present result is 30.755 aJ/~, 2, whereas the value of Kobayashi and Suzuki [7] is 12.670 aJ/,~ 2, and that of MTL [17] is 1.105 a J / A 2. The differences between the results of the present work and others may be caused by the fact that we used the observed higher excited vibrational band origins observed as the input data points, and by the fact that an SCF-CI procedure was taken and the exact vibrational Hamiltonian was used to solve the vibrational problems. All these may influence the anharmonicity of the force field. In Table 3, the 60 observed vibrational band origins (of the £ states) are listed together with the residuals ( o b s e r v e d - calculated) obtained in the final fittings, together with the results obtained using the DVR3D procedure of Tennyson [19] with the sextic force field of Ref. [8]. In the DVR3D calculation, the basis set included 50 spherical harmonics as the basis for the bending vibrations and 30 vibrational eigenfunctions, which were Morse-type vibrators with parameters D e = 0.120 hartree, t o = 0.008 hartree, rle = 1.1273 A and r2e = 1.1851 A. 1 600 three-dimensional basis functions were used in the final diagonalization of the vibrational Hamiltonian matrix. It can be seen from this table that the refined PES of this work can reproduce well the 60 observed

G. Yan et al. / Chemical Physics Letters 271 (19971 157-162

vibrational band origins up to 15000 cm -~. The corresponding root-mean-square (RMS) error is 0.34 cm ~. All the calculated band origins are within 1.0 cm t of the observed values. The maximum deviation of the calculated results from the observed values is 0.98 c m - ~ for the vibrational state (0 4 4). On the other hand, the other sextic force field gives larger deviations from the observed values even at low-lying vibrational states. In order to test the PES obtained in the final fittings, we also used the SCF-CI method to calculate the rovibrational energy levels of N20 in the ~ and H vibrational states with J = 1 and 2. The effective rotational constants B, and the vibrational energy levels for the II states are then derived from these rovibrational levels using the following approximation:

E(v,v 3, J)=

Bv[J(J+ 1)-t 2] (8)

161

Table 4 Part of the vibrational energy levels and the effective rotational constants of N_~O for the II states ( c m - t ) (u, .v2 ~'3) Uobs [7]

yeat

Uob~ -- u,.d. B,,l~al.) B,,~ob~.l

(0 1 0) (0 3 01 (1 I 0) (01 11 (0 5 0) (1 3 0) (2 1 0) ( 0 3 1) (1 1 1) (1 5 0 ) (2 3 0) (3 10) (0 1 2) (1 3 1) (2 5 0) (3 3 0) (4 1 0) (31 1) RMS error

588.76 1749.36 1880.33 2798.59 2898.28 3046.90 3166.03 3931.95 4062.20 4198.69 4336.66 4446.73 4977.99 5201.53 5490.33 5618.49 5723.42 6571.09

0.01 -0.29 -0.06 -0.30 -0.47 -0.69 -0.18 -0.70 -0.22 -0.73 -0.86 -0.35 -0.29 -0.75 -0.71 -0.72 -(I.60 -0.32 0.54

588.77 1749.07 1880.27 2798.29 2897.81 3046.21 3165.85 3931.25 4061.98 4197.96 4335.80 4446.38 4977.70 5200.78 5489.62 5617.77 5722.82 6570.77

0.4195 0.4203 0.4181 0.4158 0.4210 0.4188 0.4168 0.4167 0.4144 0.4194 0.4173 0.4157 0.4121 0.4151 0.4177 0.4157 0.4148 0.4119

0.4195735 0.4203312 0.4179180 0.4161623 0.4209484 0.4186114 0.4163822 0.4169985 0.4145010 0.4190931 0.4168060 0.4150206 0.4127326 0.4152218 0.4172133 0.4150115 0.4138621 0.4116045

and B~= [ E ( u , v ~ u 3, J ) - E ( v l u ~ u 3, J -

l)]/2J,

(9) where E(vj v~ u 3, J ) and E(~,j ~,~ t,3, J - 1) are the calculated rovibrational energy levels in the vibrational state (l, l V~U3) with the rotational quantum number J and ( J - 1 ) , respectively, G(ulv~u 3) is the corresponding vibrational energy level. For the states, l = 0, and for the II states, 1 = 1. The calculated values of B~ for the ~ states are also listed in Table 3, and t h e G ( v l v ~ v 31 and B, values for the H states are listed in Table 4, together with the values obtained from the observed rovibrational spectrum (which are taken from Ref. [3], or derived from the data appearing in Ref. [7]). One can see from Table 4 that the calculated vibrational band origins for the II state are all within 1.0 cm-~ of the observed ones. This shows that the PES obtained from the observed 2£ state vibrational band origins reproduces the II state vibrational energy levels well. In Tables 3 and 4, all the deviations of the B~ values from the experimentally obtained ones are of the order of 10-4~

10-3 cm -I

This application shows that the SCF-CI procedure is useful in the refinement of potential energy functions from observed vibrational band origins. In this

work, the PES thus obtained can reproduce the vibrational band origins for the ~ and 11 states and the effective rotational constants Bv well.

Acknowledgements Project 29673029 was supported by the National Natural Science Foundation of China. We would like to thank most gratefully Prof. J. Tennyson of London university for the DVR3D program.

References [1] G. Herzberg, L. Herzberg, J. Chem. Phys. 18 (19501 1551. [2] C. Amiot, G. Guelachvili, J. Mol. Spectrosc. 51 (1974) 475. [3] A. Campargue, D. Permogorov, M. Bach, M.A. Temsamani, J.V. Avwera, M. Herman, M. Fujii, J. Chem. Phys. 103 (1995) 5931. [4] I. Suzuki, J. Mol. Spectrosc. 32 (19691 54. [5] A. ChEdin, C. Amiot, Z. Cihla, J. Mol. Spectrosc. 63 (19761 348. [6] M. Lacy, D.H. Whiffen, Mol. Phys. 45 (19821 241. [7] M. Kobayashi, 1. Suzuki, J. Mol. Spectrosc. 125 (1987) 24. [8] J.-L. Teflo, A. Ch~din, J. Mol. Spectrosc. 135 (1989) 389.

162 [9] [10] [I 1] [12] [13] [14] [15]

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P. Jensen, W.P. Kraemer, J. Mol. Spectrosc. 129 (1988) 216. D. Xie, G. Yah, Science in China, (Series B) 39 (1996) 439. D. Xie, G. Yan, Chem. Phys. Lett. 248 (1996) 409, D. Xie, G. Yan, Mol. Phys. 88 (1996) 1349. S. Carter, N.C. Handy, Mol. Phys. 57 (1986) 175. Z. Bali6, J.C. Light, Ann. Rev. Phys. Chem. 40 (1989) 469. T.R. Cuthbert Jr., Optimization using personal computers, Wiley, New York, 1986.

[16] B.R. Johnson, J. Chem. Phys. 67 (1977) 4086. [17] J.M.L. Martin, P.R. Taylor, T.J. Lee, Chem. Phys. Lett. 205 (1993) 535. [18] W.D. Allen, Y. Yamaguchi, A.G. Cs~szhr, D.A. Clabo Jr., R.B. Remington, H.F. Schaefer III, Chem. Phys. 145 (1990) 427. [19] J. Tennyson, J.R. Henderson, N.G. Fulton, Comput. Phys. Commun. 86 (1995) 175.