Semiempirical potential curves of N2+ and CO+ computed with the MS-Xα theory

Journal of Molecular Structure (Theochem), 150 (1987) 215-221 Elsevler Science Publishers B V , Amsterdam - Printed m The Netherlands

SEMIEMPIRICAL POTENTIAL CURVES OF N; AND CO+ COMPUTED WITH THE MS-Xol THEORY*

P SENN Chemistry

Department,

ETH-Zentrum,

CH-8092 Zurwh (Switzerland)

F A GRIMM Department (USA)

of Chemistry,

The Unwers#ty

of Tennessee,

Knoxvdle,

TN 37996-1600

(Received 29 July 1986)

ABSTRACT Potential energy curves of N: and CO+ have been obtained by mapping vertical lomzatlon potentials onto the potential energy curves of the correspondmg parent species The vertical lomzatlon potentials used m this mapping procedure were computed using the MS-Xor formahsm m the muffin-tm approxlmatlon The mapping procedure reduces the error inherent m the direct use of total statlstlcal energies wlthm the muffin-tin approxlmation INTRODUCTION

The widely used multiple-scattering-Xa! (MS-Xa) method [l] has been shown to give fatly accurate results for both energies of excltatlon and energies of ionization for practically all types of molecules [2, 31. In most apphcatlons, the MS-Xor method 1s used m the muffin-tm (MT) approxlmatlon. However, the total statlstlcal energies computed mth the MS-X& method have repeatedly been shown to be inadequate for conformational analysis and the computation of spectroscopic constants. The observation by Connolly and Sabm [4] that this method predicts a linear arrangement of the nuclei as the most stable conformation of H20 has resulted m a considerable loss of credlblllty with respect to data related directly to total energies obtained with this method The reason why total energies computed with the MS-X@ method are maccurate has been known for quite some tune Calculations with correction terms aimed at ehmmatmg the errors mtroduced by the MT approxunatlon have demonstrated that the MT approxlmatlon is the maJor cause for the poor quality of potential surfaces computed wlthm the MS-Xa formalism [5, 61. According to Danese and Connolly [5, 7, 81, the total *Work supported by The Umverslty 37996-1600, U SA 0166-1280/87/$03

50

of Tennessee Computing

Center, Knoxville, TN

0 1987 Elsevler Science Publishers B V

216

non-MT energy IS obtamed approximation as follows E stat =EMT+

from

EMMT,the

energy

xjApAV&Edr+JApAVc(p)dr

computed

m the MT

- (p)4’3)dr

k

(1) The second and third terms on the right--hand side of the expression are related to the nuclear-electron and the coulomb part m the expresslon for the total statlstlcal energy. The summation 1s over the nuclei m the molecule The quantity p is the charge density which 1s obtained from the molecular orbltals, uz, as follows P = c Pf = z I1,u,u:

(2)

where the summation IS over the molecular orbltals whose occupancies denoted by n,. The quantity Ap m eqn. (1) 1s the difference between actual and the MT charge density 6, i.e.,

are the

Ap=p-p

(3)

I

I

The Xa! method 1s most widely used for the computation of energies of excltatlon and energies of lonlzatlon using the so-called “transltlon state” [9]. These energies of excitation computed at suitably sampled geometries can be used for the mapping of a potential energy surface onto the known potential energy surface of a reference state, usually the ground state. This has already been achieved successfully by Barrow et al [lo]. However, then vanatlonal technique circumvents the MT approxlmatlon Based on the formula of eqn. (I), we show that the MT approxlmatlon should give rise to smaller errors for potential energy surfaces computed with the mapping procedure than for potent& energy surfaces computed from scratch, i.e., using Es,, and spectroscopic constants computed for N: and CO+ which are m reasonable agreement with experiment are presented. In order to obtam from eqn. (1) the non-MT orbital energies, E,, we snnply use the followmg relatlonshlp between the orbital energies and the total statistical energy [ 1 l] E, =

a&atlan,

(4)

which, used m eqn. (l), gives the followmg E, = (E,)~~ + Es Ap,A V!&dr + jAPrAV&)dr k l/3

+JApAV&,)dr-

3a f 0

1

[PIP 1’3 -

Pr(ji)“3]

dr

(5)

The above equation IS fairly snmlar to eqn. (1). It can be obtained from eqn. (1) by replacmg the non-MT and the MT total statistIcal energy by the

217

correspondmg orbital energies and by replacing the total charge density by the charge densities from the z-th molecular orbital. From eqn. (2), it is readily apparent that m reasonably “large” molecules the charge densities of the molecular orbit& are a fraction of the total charge density and the MT error of the orbital energies would then be expected to be correspondmgly smaller than the MT error m the total statistical energy. As far as the mapping procedure for the computation of potential energy surfaces is concerned, we would also expect that the effect of the MT approximation would be reduced, because m the transition state approximation the required energy separations are obtained from one or two orbital energies, depending on whether an ionization or an excitation is being considered. COMPUTATIONAL

DETAILS

MS-XCYcalculations have been performed for ionizations from the three highest levels of Nz and CO The vertical lomzatlon potentials (VIPs) were computed at 10-13 different internuclear separations and then approximated m a least-squares analysis by (a) the separation between two dislocated harmonic potentials and (b) the separation between two dislocated Morse potentials The MS-Xa calculatzons The so-called “latter tail” [ 121 modification of the potential has been used. The “latter tall” consists of an analytic contmuatlon of the potential m the outer sphere region for large values of r. This modification has only minor effects on the orbital energies and is not considered to be an essential feature m this work. The atomic spheres were not allowed to overlap. In CO, the atomic radu were chosen such that the internuclear separation was partitioned among them m the ratio of the covalent radn of C and 0 [ 131. rc = (1 15/2 13)rco

@a)

r. = (0.98/2.13)rco

(6b) The Slater exchange parameters* were taken from a tabulation by Schwarz [ 141. In carbon monoxide, the Slater exchange parameter outside the atomic spheres was taken as the arithmetic mean of the values m the two atomic spheres. The computation of the spectroscopic constants In the Born-Oppenheimer by the vector r is equal

approximation, to the separation

*The Slater exchange parameters HF energy were used m this work

obtained

the VIP at a geometry specified between the potential energy

by adJusting

the total statlstlcal

energy

to the

218

surface of the Ionized product and the energy computed for the geometry speclfled by r. W)

= V,,(r) - L-d(r)

surface of the parent

species (7)

In the present case r has only one component and we consider it a scalar quantity. Let us assume that V,, and V,, are both harmonic, i.e., that V&r)

1 = T, + 3 k’(r - r;)’

and

bdr)

r

+

k”@ -

432

(8b)

where T, is the energy separation between the mmuna of V,, and V,,. From the above we obtam for the separation between the potential energy curves IP(r) = co + cl(r - ri) + c2(r - ri)’

(9)

where k’ = 2c.2 + k” rk =

(loa)

ri - EL

(lob)

k’

T, = co -ik’(ri

- rz)’

(1Oc)

The expansion coefficients co, cl, and c2 m eqn. (9) have been determined by a nonwelghted linear least-squares fit to the VIPs computed with the MSXa method as described m the previous section. A sunl1a.r approxnnatlon of the VIPs by a pair of Morse potent& gave rise to a non-hnear least-squares problem. This problem was solved by a multldlmenaonal search procedure, which, starting with several different reasonable mltlal choices gave Identical sets of converged spectroscopic constants. From the parameters defmmg the Morse potentials V,,(r) V,,(r)

= T, + D’(1 - e-@‘C’)2 = D”( 1 - e+““‘)2

(lla) (lib)

where g’ = r - rk and .$” = r - ri, the harmonic constant and the first anharmomclty can be obtained usmg the followmg exact relatlonshlps

We=P 2_lJ2 RC ( 2E.(>

o,x,

= hcw:/4D

(12) (13)

219 TABLE 1 Vertical lomzatlon potentials of N, computed with Slater’s transition state X N,(X’B+g) + N;(X%+,) A N,(X’B+~) + N;(AZnur)

+ e+ e-

B N,(X’B+~) + N;(B%+,,) + e-

X (w)”

A (w)”

B try)”

-0 9606 -1 0000 -1 0318 -10578 -10622 -10781 -1 0932 -1 1023 -1 1059 -1 1014 -10911

-17156 -1 5685 -14525 -1 3604 -1 3447 -1 2867 -12275 -1 1790 -1 1271 -10857 -10524 -10245 -0 9899

-11211 -1 1999 -1 2767 -1 3481 -13615 -14122 -14693 -15187 -15739 -16175 -16512

laoI

rNN

16 175 19 2 05 2 08 22 2 35 25 27 29 31 33 36

aX, A and B refer to the photoionization processes shown m the heading of the Table The VIPs are expressed m Rydbergs (1 ry = 109 737 cm-‘)

TABLE 2 Vertical lonlzatlon potentials of CO computed with Slater’s transitIon state X CO(X’Z+) + CO+(X%+) + eA CO(X’ZZ+) -+ CO+(AW,)

+ e-

B CO(X’Z+) + CO+(B*x+) + erc0

(4

16 175 19 2 05b 2 13 22 2 35 25 27 29 31 33 36

X (ry)” -0 -0 -0

8885 9092 9324

-0 9711 -0 9831 -10074 -1 0282 -1 0458 -10560 -1 0548

A O-Y)” -17511 -16141 -15038 -14122 -13697 -1 3356 -12706 -1 2156 -1 1541 -1 1038 -10623 -10279 -0 9860

B O-Y)” -1 2172 -13045 -13636 -1.3993 -14111 -1 4187 -14291 -14361 -14462 -14602 -14753

aX, A and B refer to the photolomzatlon processes shown m the heading of the Table The VIPs are expressed m Rydbergs (1 ry = 109 737 cm-‘) bThe SCF iterations did not converge for the transltlon state corresponding to the photolomzatlon of the hlghesi level

220 PRESENTATION

AND DISCUSSION OF THE RESULTS

The computed VIPs are shown m Table 1 and m Table 2 The computed spectroscopic constants are shown m Table 3. In the Morse approxlmatlon, the differences between computed and experimental equlhbrmm internuclear separations range from 0.027 to 0.132 a, m magnitude. Gunnarsson et al. [15] did some Xa calculations with non-MT corrections included They computed r, to an accuracy of 0.1-O 2 a0 and the harmonic constants to an accuracy of 100-200 cm-‘. The experimental equlhbrmm internuclear separations are reproduced m this work with an accuracy which would be expected from non-MT Xor calculations, but the same 1s not the case for the harmonic constants However, the harmonic constants computed m the harmonic approxunatlon differ considerably from the ones obtained with the Morse approxlmatlon This 1s not the case for r, computed with the two TABLE 3 Computed and experimental spectroscopic constants r of Ni and CO+ Molec ion

Harmomc

Morse

Experiment

N;A’n:, N+B*Z+ 1 u co+xzz+ CO+A*n, CO+B28+

1 1 1 0 1 1

056 326 323 972 333 390

1059 1303 1 328 0 969 1315 1 405

1 145 1 228 1 377 1030 1219 1 448

6 (a,)

N+X%+ N;A’n ‘u, N+B?Z+ cb+xzi+ CO+A*n, CO+B*B+

2 019 2 236 1910 2 069 2 321 2 035

2 021 2 271 1946 2.066 2 381 2 077

2 110 2 220 2 030 2.107 2 350 2 209

wk (cm-‘)

N+X%+ NZfA 71 tt N+B%+ &+x4+ CO+A*n, CO+BZz+

w exk (cm-l)

N+X*x+ N;A’n:z N+B%+ &+x4+ CO+AZn, CO+BZs+

T, WI

N+X%+

2261 2518 2237 2114 2331 2026

2482 1831 2902 2397 1549 2004 17 15 14 15 15 11

2207 1904 2420 2214 1562 1734 16 15 23 15 14 28

aThe spectroscopic constants b& and w,x; are m wavenumbers (cm-* ) and T, 1sexpressed m Rydbergs (1 ry = 109 737 cm-‘) The equlhbrmm mternuclear separations are expressed m Bohr

221

methods. It 1stherefore possible that a slightly better agreement with experlment would result from more accurate representations of the potential energy curves In large scale CI computations, spectroscopic constants of dlatomlc molecules can be computed with considerably greater accuracy than the accuracy which has been achieved m this work [ 161 Although the results of this work are not lmpresslve m terms of their agreement with expernnent, they do not seem to reflect the usual grave consequences resultmg from the MT approxlmatlon when computmg potential energy surfaces. Because of the avalablhty of MS-Xa programs, these results suggest that the MS-Xcz method when used with the proposed mapping procedure could be useful m obtaining crude potential surfaces for large molecules where more sophlstlcated methods are too costly or unavailable. REFERENCES 1 J C Slater, Adv Quantum Chem , 1 (1964) 35 2 K H Johnson and F C Smith, Jr, Phys Rev., B, 5 (1972) 831 3 K H Johnson and U Wahlgren, Int J Quantum Chem , 6s (1972) 243 4 J W D Connolly and J R. Sahm, J. Chem. Phys , 56 (1972) 5529 5 J B Danese and J W D Connolly, Int J Quantum Chem , 7s (1973) 279 6 J P Worth, B I. Dunlap and S B Trlckey, Chem. Phys Lett , 55 (1978) 168 7 J B Danese and J W D Connolly, J Chem Phys ,61(1974) 3063 8 J. B Danese, J. Chem Phys , 61 (1974) 3071 9 J C Slater and J H. Wood, Int J Quantum Chem., 4s (1971) 3 10 W L. Barrow, Hldeo Samhe and R. H Felton, Chem Phys Lett , 68 (1979) 170 11 J C Slater, Adv Quantum Chem., 6 (1972) 1 12 R Latter, Phys Rev, 99 (1955) 510 In the present work, the coulomblc long-range potential was with 2 = 1 13 J C Slater, J Chem Phys ,41 (1964) 3199 14 K Schwarz, Phys Rev B, 5 (1972) 2466, Theor Chum Acta, 34 (1974) 225. 15 0 Gunnarsson, J Hsxrls and R. 0 Jones, Int J Quantum Chem , 11s (1977) 71 16 G Theodorakopoulos, S D Peyerlmhoff and R J Buenker, Chem. Phys Lett , 81 (1981) 413