Volume 78, number 2 CHEMICAL PHYSICS LETTERS 1 March 1981

THE RELATMSTIC CORRECTION TO THE EXCITATION ENERGY OF FORMALDEHYDE

Plulip PHILLIPS and Ernest R. DAVIDSON

Deparrnlenr of Cilemisrry. Uiliversity of Washington. Seattle. Washington 98195, USA

Received 2 December 1980

Fust-order perturbation theory for the mass-velocity, Danvm, and magnetic corrections was used with a non-relativistic wavefunctlon to cakuhxte re]ativistlc conttibutlons to the ground and 3A2, 3A”, 3Ar, *Bz, and 2B1 excited states offor- maldehyde. Corrections to the evciration energy were found to be in the range of 30-80 cm-‘.

I_ Introduction

Aoyama and Yamakawa [ 1 ] recently reported a

0.3 eV relativistic correction to the 3A,-IA, energy gap in CH,O. In their calculations they used a Breit- Pauii type hamiItonian involving only the mass-velocity and spin-same orbit interaction. They reported Tamm- Dancoff (TD) and relativistic Tamrn-Dancoff (RTD) calculations. The difference between the TD and RTD results provided their estimate of the relativistic correc- tion to the 3A2-1A1 energy gap. This correction was of the same sign and magnitude as the apparent error (compared with experiment) in the best non-relativistic calculations of the excitation energy.

WhiIe this resuh is intriguing, it also seems unhkely to be reliable because the Darwin correction is expected to be of nearly equal magnitude and opposite sign to the mass-velocity term. We report here a sequence of calculations using a more complete form of the Breit- Pauli hamiltonian. These calculations show a much smaller relativistic effect than found by Aoyama and Yamakawa_

2. Method

Aoyama and Yamakawa [I] reported that essentially all of their correction came from the mass-velocity operator_ Hence we first attempted to reproduce their result by performing a coupled Hartree-Fock calcula- tion using the mass-velocity operator as a perturbation.

We also simulated their TD results by calculating the first-order correction to the excitation energy with the mass-velocity operator and a TD wavefunction.

In calculations we regard as more reliable, -we evalua- ted the relativistic effects on the excitation energy of several states of CH,O using fxst-order perturbation theory for the harmltonian H(l) =f$, f & fff3 f H4 where, in atomic units,

(mass-velocity) ,

4 = 3 ~FZAS@j~) (one-electron Darwin) ,

(two-electron Darwin) ,

(magnetic) .

We arrived at this form of W(l) by examining the Breit-Pauli hamlltonian as modified by Pyykk6 [2] and dropping terms expected to be negligible for light atoms. The reliability of this approach has been demon- strated for several states of the carbon atom in a recent

paper 131. In all calculations a Dunning [4] double zeta gaus-

siau basis set, augmented with polarization d functions on oxygen and carbon and polarization p functions on hydrogen, was used. Parent configuration restricted Hartree-Fock wavefunctions and first-order relativistic

230

Volume 78, number 2 CHEMICAL PHYSICS LETTERS 1 March 1981

corrections were generated at the flat equilibrium geom- etry of [S] of the ground state for the lAl ground state, the 3A1(~~*) state, the 3AZ(nm*) state, the 2B2(n-1) ionized state and the 2BI(n-1) ionized state. A similar calculation was performed for the 3A” energy at the Jones and Coon [6] 3A” geometry.

3. Results

In order to estimate the accuracy of this method we treated the ground state of each of the atoms of CH20 in this same approximation. Table 1 gives the contribution of each term in H(l) to the total relativistic correction. These values agree within 1.5% with the corresponding values obtained by Fraga et al. [7], Desclaux [S] , and Veillard and Clementi 191. The values for the mass-velocity and Darwin corrections individually agree within 4% with the results of FraG et al. [7].

The sum of the relativistic corrections to the atom-

Table 1 ReIativistic corrections for atomic ground statesa)

Relativistic Carbon Oxygen Hydrogen correction C3P) CP) cas>

mass-velocity -16632.7 -55009.8 -6.7 one-electron Darwin 13331.2 43459.3 5.3 two-electron Darwin -291.2 -8775 magnetic 582.3 1755.1

total -3010.3 -10673.0 -1.4

a) AU values are reported in cm-l.

ic ground states are usually assumed to provide a good approximation for the relativistic corrections to all states of a molecule. As table 2 indicates the relativistic correction to the lAl state of CH20 is -13601.5 cm-l while the sum of the atomic corrections are -13686.2 cm-l _ This corresponds to a difference of 85 cm-l between the molecule and the free atoms. If the two- electron Darwin and magnetic corrections are omitted, the relativistic correction to the LA, state of CH20 becomes -1.82 eV which agrees well with the Aoyama et al. [lo], Dirac-Fock value of -1.8 eV.

Table 2 also reports the relativistic corrections for several excited states of CH20. The two calculations, 3A2 and 3A”, have relativistic contributions to their excitation energies from lAl of -28.5 cm-l and -32 cm--L respectively. in this case the geometry de- pendence of the splitting (3.5 cm-l) is quite small. Furthermore our value of -28.5 cm-1 does not agree with the Aoyama and Yamakawa value [l] of +2150 cm-l. Our coupled Hartree-Fock and TD values for the mass-velocity contribution to the 3A2-‘A1 energy gap were -29 cm-l and -18 cm-L respectively. Thus we were not able to reproduce the values reported by Aoyama and Yamakawa.

Finally table 2 shows that the ionized states have the largest relativistic corrections_ One might have expected that removal of an electron would decrease the magnitude of the relativistic correction. But our results indicate that the relativistic correction to the 2BZ-1AI and 2BI--‘AI energy gaps. -68 and -74 cm-’ respectively, are the largest of the excitation energy corrections and are of opposite sign to that ex- pected.

Table 2 Relativistic corrections for CH,Oa)

Correction ‘Al

mass-velocity -71236.4 DarwinonC 13174.0 DanvinonO 43371.4

Darwin on 2Hs 15.3 two-electron Darwin -1074.8 magnetic 2148.6 total -136015 @ittinS with ‘A1

a) All values are reported in cm-l _

3A2 3 A..

3A1 2B2 -1

-71265.6 -71283.8 -71270.4 -7 1429.4 -71434.5 13141.6 13153.2 13150.8 13191.1 i32G8.2 43406.4 43406.6 43392.5 43480.1 43458.6

14.3 15.1 15-l 13.6 14.4 -1073.2 -1075 -4 -1073.3 -1075.2 -1077.4

2146.4 2150.8 2146.7 21503.8 21548.3 -13630.0 -13633.5 -13638.7 -13669.4 -13675.9

-28.5 -32Q -37.2 -67.9 -745

231

Volume 78, number 2 CHEkfIICAL PHYSICS JXTTERS 1 March 1981

Table 3 Is orbital expectation values for mass-velocity and Darwin termsa)

C 19, mass-velocity C Is, Darwin sumforcls 0 ls, mass-velocity 0 is, DarWln for 0 lb

free atom s AI sA2 3 A” 3Al =BZ *Bl

-7526.1 -7811.0 -7804.0 -7804.4 -7804.1 -7810.0 -7811.1 6361.4 6355.5 6350.6 6350.9 6350.7 6354.6 6355.7

-1464.8 -1455.5 -1453.4 -1453.5 -1453.4 -1455-4 -_1455.4 -25511.7 -25529.8 -25524.3 -25524.0 -25514.6 -25532.7 -25525.6

20638.7 206.50 3 20646.8 20646 5 20643 2 20652.2 20647.7 -4873.0 -4879.5 -4877.5 -4877.4 -4871.4 -4880.5 -4877.8

a) Ali values are reported in cm-‘.

4. Discussion which differ by one in their 2s orbital occupation.

As indicated in the previous section there is an 85 cm-l difference between the relativistic corrections for the free atoms and for the IA, state of CH20. This difference is not due to modification of the 1s orbitals of oxygen or carbon as table 3 shows a negli- gible difference (6 cm-l) between the total values for the free atoms and the moIccuIe.

Fig. 1 and table 5 aid in pinpointing the source of the variation in the correction between states. Oxygen atom resutts suggest a linear relationship between the oxygen 2s population and the total reIativistic correc- tion_ For the states of the nezctral molecule the slope of this Tine, -385 cm-l/eIectron, agrees well with the difference, -329 cm-l. between the corrections for the sp5 3P and s2p4 3P states of the oxygen atom

Fig. 1. The the neutral

232

The 1 Al state has a 2s population of 1.82 due to promotion in forming the molecule from the atom and hence has a reduced correction compared with the atoms. The excitations n+ or nrr* to the first approxi- mation move an electron from a p orbital on oxygen to the predominantly carbon pn* orbital and might be expected to reduce the relativistic correction. There is a back transfer of charge in the al orbitals which increases the 2s oxygen population. Further the reduc- tion of electron population near the oxygen increases the sp energy gap and reduces the extent of sp promo- tion. Both of these effects tend to increase the 2s population and, hence, the relatrvistic correction. TabIe 4 shows that the 3a, and 4al moIecular orbit& increase their oxygen component {and hence theirp4 value) while the 5al orbital shifts toward CH2 and de- creases Its p4 value,

-13700 1 1 I I 1 I I I I I I8 IS 20

OXYGEN 2s POPULATION

relativistic correction. in cm-i, for states of C&O plotted against oxygen 2s population. The least squaws Iine throw& molecule states, RC = -385 (pop) - 12917, is shown.

Volume 78, number 2 CHEMICAL PHYSICS LETTERS

Table 4 Molecular orbital expectation values for negative of mass-velocity terma)

I March 1981

MO l Al 3Aa 3A'"c)

la1 25529.8 (2)b) 25524.3 (2) 25524.0 (2) 2al 7811.0 (2) 7804.0 (2) 7804.4 (2) 3ar 1332.3 (2) 1395.0 (2) 1384.4 (2) 481 501.4 (22) 547.0 (2) 552.7 (2) 5ar 228.0 (2) 140-4 (2) 136.9 <2) lb1 77.3 (2) 105.1 (2) 115.5 (2) 2br 49.2 (0) 56.4 0) 79.3 (I) lbLz 47.1 (2) 23.3 (2) 23.8 (2) 2b2 91.3 (2) 130.5 (1) 131.1 (I)

a) All values are reported in -em-r _ b) Occupation number for each orbital. e) Orbitais labeled with C a,, symmetry wrth which they correspond.

3A1 2J%2 2B1

25514.6 (2) 25532.7 (2) 25525.6 (2) 7804.1 (2) 7810.0 (2) 7811.1 (2) 1380.4 (2) 1367-O(2) 1354.7 (2) 552 9 (2) 593.8 (2) 599.0 (2) 148.4 (2) 219.4 (2) 1845 (2) 112.0 (1) 104.6 (2) 11x.3 (1)

63.3 (1) 75.5 (2) 27.3 (2) 65.0 (2) 77.7 (2) 141.1 Cl) 91.9 (2)

The ionized states of CH20 he -50 cm--l off the linear correlation of fig. 1. The source of this deviation can be seen in the correction for O*_ As for the mole- cule, ionization of oxygen increases the magnitude of the relativistic correction. Clearly for Oi the 2s popula- tion remains equal to two. Removal of the 2p electron decreases the shieiding of 2s and leads to an increase of the 2s correction which more than compensates for the loss of a 2p electron. The relativistic correction for the 0+ s2p3 %$ state is 229 cm-l larger in magnitude than for the oxygen ground state. Table 5 shows that the oxygen population in CH20 decreases by about haIf of an eIectron as a result of ionization.

The key conclusions from these results are (1) first-

Table 5 MuJ,bken gross popukxtions for the excited states of C&Q

Orbital ‘AI aAa 3A’* 3Ar =B2 -1

c 1s 2.00 2.00 2.00 2.00 2.00 1.99 c2s 1.15 1.10 1.16 1.12 1.21 1.24 C 2Px 0.80 0.71 0.68 0.75 0.77 0.79 C+Y 1.13 1.15 1.15 1.13 1.32 1.32 C%?z 0.68 1.15 1.10 0.99 0.39 0.17 01s 2.00 2.00 2.00 2.00 2.00 2.00 029 1.82 1.84 1.84 1.85 1.86 1.86 Q2Px 1.50 156 157 155 1.53 150 0 2Py 1.88 1.02 1.03 190 1.02 191 Q 2Pz 1.30 1.82 1.82 0.99 1.59 0.82 His 0.77 0.74 0.74 0.79 0.55 0.58 H2s 0.77 0.74 0.74 0.79 0555 0.58 tot@) 15.82 15.82 15.83 15.83 14.78 14.79

a) Residual charge is in other orbitals.

order perturbation theory for H(l) produces results that agree with Dirac-Fock calculations for light atoms, (2) the geometry dependence of relativistic effects are small, (3) there is a definite correlation be- tween the oxygen 2s population and the relativistic correction, and (4) ionization of p electrons causes contraction of the s orbital and a resultant increase in the relativistic correction_

This work was partially supported by a grant from the National Science Foundation_

References

[l] T. Aoyama and H. Yamakawa, Chem. Phys. Letters 51 (1977) 508.

121 P. PyykkG, Advan. Quantum Chem. 11 C1978) 353. [3] E R. Davidson, D- Feller and P. Phillips, Chem. Pbys.

Letters 76 (1980) 416. 141 T.H. Dunning, J. Chem. Phys. 53 (1970) 2823. 15 ] S.R. Langhoff, ST. Elbert and E-R. Davidson, Intern.

J. Quantum Chem. 7 (1973) 999. [6] V.T. Jones and J-B. Coons, J. Mol. Spectry. 31 (1969)

137. [7] S. Fraga, K.M& Saxena and J3.W.N. Lo, At. Data 3

(19’71) 323. 181 J.P.Desclaux, At.DataNucl.DataTables12 (1973) 311.

19 ] A. Veillard and E. Clementi, J. Chem. Phys. 49 (1968) 2415.

[lo] T. Aoyama, H. Yamakewa and 0. Matsuoka, J. Chem. Phys. 73 (1980) 1329.

233