Volume 65. number 1 CllEXlICAL PllYSlCS LI-STTI:RS 1 Aoym 1979

ENERGY GRADIENT IN A MULTI-CONFIGURATIONAL SCF FOIWALISM

AND ITS APPLICATtON TO GEOMETRY OPTIMIZATION OF TRIMFI-HYLENE DIRADICALS

Shigcki KATO 3rd Keiji hlOROKUhlA

lrrrrirurr for .~ofccuiar Scicrtrr. Alpxhiji. Okaxki -J-I-I. Japan

Rcccircd 19 \I~rch 1979

Prc\iour methods of analytic eahtation oC ths potential cncg? pltxiient with tcspcct to nuclear axwdinxcs hate been limited to sinslc dctcrminant llxtte-I-o& xtrcfunctions. An c\pr&on and a practical mcthnd arc prcslrntcd fnr ana- iylic gralimr crdu;ltion for MCSCI’ t\avcfunctionr of twtain witlcly applktblc t)pcs A gwtnctry optintilation of tri: mcthylcnc diradial rhction interntediatcs has been xtu~ll~ accontplirhrd by the use 01’ the snrrgk’ pdicnt for an ab tm- tio MC SCI’ w3vcfunction.

1. Introduction

In order 10 understand IlIe niecktnistti of chtniic31 re3crions. it is cssentiztl to h3ve knowled_ce of some critic31

points ott the potenti etlcrgy surf3ce [I 1, such 3s :hc re3cr3nt. rhc producr. the Ir3nsitiot1 st3tc [Z-3] 3nd the

rcmion intcrnwdiate. Recently an cfftcient tvctltod kts been dewlopcd IO IOCXI~~~LI equilibritm geotttcrrics and I~WI-

sition slate structures of polyatotnic systc~m [?.S] Contbitted witlt 3n3lytic cv3lu3tiot1 of the gr3diet11 of the pu- tr’uli31 euergy with rcspc‘ct to the nuclar coordir13tcs [G-O]. thcorctic~l cl13r3ctctir3tirx1 of cquilihriun1 111olecul3r geomcrrics 3nd (r3t1sitiun state structures 113s beet1 c3rriccl out using 3b initio X10 111~1htxls [S] _ TIE et1crgy grdi-

em method 113s 31s~ been used 10 tr3ce rhe entire reaction patIt [ 101 dcfined mzcording IO rhe inrrinsic rcmion

ctxmlin~tc (IRC) formalism [ l I]. The cncrey gr3dict1l 3ppro3cl1 is 3lre3dy proving to be 3 most parcrful tooI for tl1corclic31 st1ttJics of mokcul3r

structurs 3rd rc3ctivity. Iis most serious deficis11cy 113s bstxt its praclicd litttitation to sitt_clc-Jctertttittantal

lhrtrcc -Fock (II!-) \v;1vcfunctions. This precludes its 3pplic3tion 10 t113ny chct11ic3lly signific3t1t processes. l’or cx3mplc. the IIF w3wfut1ction provides 3 clu3lit3rivcly incorrect dcscriptiot1 of the clcctrottic structurr’s of dir3di-

cd species which 3ppc3r in molccul3r dissociation processes 3nd in scrt3in rypcs of re3ction intcrmcdi3rcs. It 13ikS

cssenti31 correl3tion effects which cl13r3cterize clectroaic st3tes it1 wlticlt old bonds 3rc \\~\3k~n~tl or IICU body

arc bcittg fortttcd. Our purpose here is 10 ovcrcot11c this difficulty. III thr present p3prr \kc present an efficient 111etl1otl for c3lcul3ting lhr cnrr=_ _ * ov cr3diCt1t it1 tl1a I11ulti_cot15~ur3-

lion31 (MC) SCF tlicory [I?. 131. \\‘e also 3ppIy tlic nicthod 10 tltr: geo111rtry optin1irxion of tritt1cth~ Ir‘tlt ICX-

tion it1tsrnwdi3tes. To our knowledge this is the first 3n3lytic ev3lu3tion of 111 c energy pr3dient bcynnd tl1c III’

theory.

2. Method of calculation

Thaugl1 fort1131 esprcssions of rhc energy gr3dienr for the a * netwr31 SCF theory l13ve bcct1 gi\cn prc\iottsly [ l4.6] no explicit espressiou suitable for comput3tiot1 is 3\3il3ble in the literztturc. So NY start wirlt 3 brief dcriv3tim1.

In the hfC SCF theory. the elecrronic waveiunction. *, is cspresscd 3s 3 linesr cotnbin3tion 01~ config11ra:ion

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Volume 65, number I CHEMICAL PHYSICS LETIEFtS 1 August 1979

functions, Cgy,

\k = CAfbj Z

(1)

and the configuration functions are constructed by the appropriate combination of Slater determinants composed of a set of orthonorma1 molecular spin orbitak ~j~ and ~j~- The variational condition for the total electronic energy with the constraint of orthonormality of MO’s leads to the following equations:

t-i.4 = AE, 0)

(3

Eq_ (2) is the usual configuration interaction (%I) secular equation to determine the coefficients for the config- umtion expansion_ The orbital ser is determined by soIving the general SCF equation (3), where Fi is the Fock operator for the ith orbital and eii is the Lagrange mu!tiplier_ in the LCAO theory:

eq. (3) is reduced to the optimivtion of MO coefficient Cri_ Eqs_ (2) and (3) are simultaneously soIved and the stationary condition is achieved with respect to both the Cl coefficients.Az. and the MO coefficients, C,i-

The totaI eIectronic energy, E, in MC SCF theory is given by [ 151

(5)

where subscripts i, j, k and I denote MO’s and r, s, t and u denote AOS. The first term of eq. (5) is the one-electron

part including both kinetic energy and nuclear attraction contributions_ The two-electron part is given by the sec- ond term; and the nucIcar repulsicn energy is the third term. The vector coupling coefficients, -Q and ‘;i- RI, are functions of the CI coefficients dererminrd in eq. (2). The derivative of the ener>y with respect to a nu&ar coor- dinate. (I, is given by

The first and the second terms result from the variation of CI coefficients, A,. accompanying a nuclear displace- ment_ It can e@Iy be shown that these two terms vanish if we introduce the variationa condition given by eq. (2).

By taking into account the variational condition for the MC SCF orbit4 equation [ 151,

<iIF- - FiIj> = 0 (7)

and the orthonormality condition of MO’s, we can easily show that the third and the fourth terms of eq. (6) may be rewritten with the use of Lagrange multipiiers given by eq. (3) as follows,

Volume 65, number 1 CHEMICAL PHYSICS LJ3TERS 1 Augwst 1979

09

This replacement is allowed only when the total wavefunction Yr is optimized for the MO coefficients as in the MC SCF method, but is not allowed for the CI wavefunction in which &IO’s are not optimized_ Thus the final eh- pression for the gradient is given by

(9)

The cakulation of the ener,v gradient by the use of eq. (9) is expected to be rather time-consuming. One has to evaluate n4/S elements of the second order density matrix, where II is the number of AO’s. It is, therefore, desirable to find a simpler form of the gradient expression which allows a faster gradient evaluation. If configura- tions included in the MC SCF wavefunction are limited to pair type two-. four-_ six-, ___ electron excitations from the orlginally doubly occupied orbitals to the virtual orbitals. the total energy can be expressed [ 161 by

wvheref;: is the occupation number of the ith MO, and Jii and A;; are the Coulomb and the eschange integral, re- spectively. The vector coupling coefficients for the Coulomb and the exchange integrals are given by aij and bii The generalized valence-bond (GVB) method [I 7, IS] and various types of open shell SCF theories [ 191 have the same energy expression_ Although the MC SCF method with the energy expression given by eq_ (IO) is limited in the configuration selection, the essential part of the electron correlation effect which is important in describing the breaking and formation of chemical bonds in a polyatomic system is included in this form. For example. the molecular dissociation processes and the non-concerted bond-interchange can be treated using this type of RIG SCF wavefunction

The energy gradient for eq_ (10) is written as follows:

where PLY is the number of core orbitals (whose occupation number is 2) and the JL, is the number of valence orbit- 31s with variable occupation number_ Open she!1 orbitals are included in the valence orbitals. The density ma?rices for the tore, 0:“. and the valence orbit& DA_ are given by

Ire 0~” = C C,i C,i , Do* = Cri C,i - (13

i

The vector coupling coefficients between the core orbital and the ith valence orbital are given by nio_ noi b,, and hoi in eq_ (1 I)_ This new MC SCF gradient expression allows one to use nearly the same algorithm as in the single detemlinant HF method.

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Volume 65, number I CIiEMICAL PHYSICS LETTERS I Aeust 1979

We have written a program to calculate the enerm, eq. (IO), and its gradient, eq. (1 I), in the MC SCF theory which is also applicabIe to the GVB and the open-she11 theories_ It has been implemented in the IMS ab initio package (GAUSSIAN 70 [20] + HONDO [21] f own gradient and optimization programs 19, lo] + many other own routines)_ The two-by-two rotation procedure was used to solve the MC SCF equation [ I3]_ The increase in the computer time for the energy gradient going from the HF method to the MC SCF method was found to be negligible_

3_ Geometry optimization of trimethyIene diradical intermediates

The singict trimethykne’diradical is an important reaction intemlediate invoked in the geometrical isomeriza- tion of cyclopropane [22]_ Three possible forms of trimethylene, that is, the edge-to-edge (EE). the edge-to-face (EF). and the face-to-face (FF) trimethyiene, have been proposed [3-S] _ Since trimethylene has two unpaired elec- trons, the single determkant HF approximation does not give a correct description of the eIectronic structure of this species; the MC SCF method removes the defect of the HF method. Previousiy some ab initio hi0 calcuiations have been performed on trimeth> lene :24--26]_ A semi-empirical (MINDO/3) g eometry optimization has also been carried out within the singlet unrestricted HF(UHF) approximation [??I_

We have performed S-configuration MC SCF gradient calcuhtions with the split-valence 4-31G basis set to de- termine rhe optimized geometries of the above mentioned zhree forms of trimethylene. Since the geometry iso- merization of cyclopropane is characterized by breaking a CC bond and subsequent rotation of CH2 groups, the configuration in which tile two eIectrons are excited from the bonding orbital (u,) Iocalized in tfle breaking CC bond to the corresponding antibonding orbital (u;) is important_ Furthermore. another occupied orbital (u?) which is degenerate with the breaking CC bond orbital at the cyclopropane geometry should be correIated with its anti- bonding counterpart (~5) in order to maintain the D jl, symmetry of the cyclopropane wavefunction- Four relevant orbit& in the EF form are schematically illustrated in the following:

Thus. the following five configurations involving excitations between these four orbitals are used to describe the geometricat isomerir3tion processes of cyclopropane

+I =corc-ufuz, *f*z = core -(ur)‘us , 4r5 = core - uf (~5)~ _

+4-Z core -vz(u;)2 . ;1B5 = core - (@u$ _ (13)

For all the geometries studied, +l and 9, make an almost 50:50 contribution whereas the weights of the other configurations are found to be Iess th&i cfew prrcen:_ The geometry optimiation procedure we used is as follows: (1 )For EE and FF trimethylenes zlt first the singlet UHF geometry optimization was performed and the resultant geometries were used as ;he starting points for further optimization in rile MC SCF method and (2) for EF trime- thylene MC SCF geometry_optiuCzation was started from the point at which one of the CH, groups WJS rotated by 90” in the EE optimized geometry_

The optimirtrd geometries of EF and EE trimethylenes are shown in fig_ I_ Their energies are a!so shown in the figure. The energy of EF trimetll lent is tower than that of EE trimethyIene by about 1 .O kcal/mol_ (A preliminary 2-configuration !W SCF geometry optimization with the STO-3G basis set puts EF lower than EE by 02 kcal/mol_) These resuits are different from those of Horsiey et al. [25] and Salem [24], who have used a minimal ST0 basis set and an HF method supplemented \+ith a 3 X 3 CI and have found that the EF form is higher in energy than the

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Volume 65. number 1 CHEMICAL PHYSICS LE-I-I-ERS 1 Actcgst 1979

EF(Cs 1 0.3 k-4

E=-~~GBKx)

E--11632@&

Fig_ I_ Optimized geometries and enerE,ies of El’ _md 1-X tri- meth, lens dimdkdr in a S-cnnfiguration MC SCI- -xith the 4-31 basis set. Energ& are in hartree. band distawes in -1 and bond angles in degrees.

EE form by about I.6 kd/m01. A GVB calc~l31i0n gixs similar results [261_ Though our calcularion is superior

to these in rhe basis set, the wavefunction (hfC SCF) and the geometry optimizarron, rhe energy difierenct be- tween EE and EF is probably too small to be conclusive_ The question whether these oprimired gsornetries corre- spond to local rnininw or transition states should be determined by the normal coordinate analysis af these points. which WC have not done_ However a felt iterations of geometry optimization \\ith a broken symmetry suggest that EF is a transition state and EE is a local minimum_ This characterization is in agreement with Salr~n‘s. in which EF is a transition state for opticlal isomerization and EF is an intermediate for geometrical isomrriz~iion. If EF is actually Iower in energy than EE, as might be su ggested from our results, optical isomerization shouid take place more easily than geometrical isomerization. This is in accord with Berson’s experimen:al findmgs with deu;erocyclopropane [ZS] _

As shown in fig_ I, the central CCCangles of EF and EE trimethylene rare nearly identic?-l_ The only geontetry

w.z can compare with this is HorsIey’s transition state \\hich seems to be located not esacriy a1 but \cry close to EF trimethylene_ Their geometry is found to be very similar to that of our EF transition state.

There was 110 local minimum on the kfC SCF potentiai energy surface corresponding to FF trimethylene: the MC SCF geometry optimization has ted to the equilibrium c: *-rometry of c> clopropane. This result agrees v.ith rhe previous ab initio calculations_ On the other hand. in our corresponding UHF calcuktions there exists an FF imrr- mediate, which is also found in the semi-empirical UHF calculation [27] _ This discrepancy is attributable io rhe inclusiou of states with a higher spiu multiplicity caused by the breakdown of the spiu synimtry in the UHF method_ The wavefunction of a diradical electronic state is expressed in the UHF approximAion by 3 linear COIW

bination of singlet and tripIet spin functions, and consequently the singlet potential surface contLtins a mirae of the triplet energy surface. When the triplet surface hasa iocal minimum, the singlet UHF may exhibit a false min- imum, which must be the case here. Thus, a spm-adapted v,avefunction such as the MC SCF wavefunction. is re- quued to describe a reliable potential surface for a diradical electronic state_

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Volume 65, number 1 CHEMICAL PHYSICS LETTERS I August 1979

4_ Concluding remarks

We have demonstrated in the present paper that for MC SCF wavefunctions of certain widely applicable types (including GVB wavefunctions) the energy gradient with respect to nuclear coordinates can be calculated analyti- caiIy with onIy a small increase in computationat effort over the HF energy gradient. We have applied the MC SCFgradient method to geometry optimization of intermediates and transition states of cycIopropane iso- merization. The HF method is unable to describe this diradica1 species even qualitatively and the UHF method is not ahvays reliable, as was shown above_ The energy gradient with CI wavefunctions is difficult to evaluate. The hlC SCF gradient method is probabIy the only efficient and reiiable way to optimize the geometry and to deter- mine the energy of such a system_

Thus the appIic%tion of the MC SCF gradient method to studies of potentiai energy surfaces for chemical reac- tions appears to be very promising_ We are using the method to determine the geometry and relative energy ofre- actants, products, transition states and reaction intermediates. We aIso determine the normal coordinates at these points and foilow paths of reaction aIong IRC. Results of such calculations will be reported elsewhere.

Acknowledgement

The authors are gratefu1 to Professor H-F. King and Dr. K_ Kitaura of this Institute for helpful discussions and comments_ NumericaI calculations have been carried out at the Computer Center of this Institute on a HITAC M-l SO computer.

References

[ 11 1I.F. Schaefer III, The electronic structure of .ttomsand moierules (Addison-We&y, Reading, 1972)- [7-I S_ G&stone. K J_ LaidIer And Ii_ Ewing. Theory of rate processes (hfcGraw-Hill, %x York, I94 I)_ (31 R-A_ Marcusand 0.K Rice. J. Phbs. Coiloid Chem_ 55 (1951) 894;

R-A_ Marcus. J. Chem_ Phys_ 20 (1952) 359;43 (1965) 26.58. [Sl J-K’_ Slcher Jr_ and A. KomornicIii. Chem_ Phys_ Letters 10 (1971) 303_ [51 J-W’_ hfciver Jr. and A_ Komomicki. J_ Am_ Chem_ Sot_ 94 (1972) 262% 161 P_ Pufay. inr afodem theoretical chemistry, Vol_ 4. ed. H-F_ Schaefer III (Plenum Press. New York, 1977) p_ 153. 171 J. Germtt and 1.X MdIs, J. Chem. Phys. 49 (1968) 1719. [ 81 R_ Moccia. Theoret_ aim_ Acta 8 (1967) 8. [PI A_ Romornicki, K_ ishi&, K_ hforokuma, R_ Ditchfield and M_ Conrad. Chem. Ph>s. Letters 45 (1977) .595-

[IO] Ii_ ishida. R_ Slorokumn and K_ b!omornicki, J_ Chem_ Ph,s_ 66 (1977) 2153; B.D. Joshi and Ii_ &forokuma. J_ Chem_ Phys_ 67 (1977) 4880.

[ 1 fj K_ Fukui, J_ Ph>s_ Chem- 74 (1970) 416i; K_ Fukui. S_ i&to and Ii. Fujimoto, J. Am. Chem. Sot. 97 (1975) I; S. Kate and K. Fukui, J_ Am_ Chem_ Sot_ 98 (1976) 639.5

[I21 A-C. Wahl and G. Das, in: Modem theoretiuI chemistry. Voi_ 3. cd_ H-F_ Schaefer Iii (Plenum Press. New York. 1977) p- 5i_ [ 131 J. Hinze, J. Chem. Phys. 59 (1973) 6424_ [ 141 31 D. Kumanova. Mol. Phys. 23 (1972) 407.

B. Levy and G. Berthirr, intern. J. Quantum Chem_ 2 (1968) 307_ A_ Veilhrd and E. Clementi. Theoret_ Chim Acta 7 (1967) L33_ W-J. Hunt, P J. Hay .md W.A. Goddard III, J_ Chem. Phys- 57 (1972) 738. F_W_ Bobro\\ia and W-A. Goddard III, in: Modern theoretial chemistr)‘_ VoL 3, cd. iI_F_ Schaefer III (Plenum Press, New York, 1977) p. 79. C_CJ_ Roothsan. Rev. Mod. Phys. 32 (1960) 179_ WJ_ Hehre. W-A_ Lathan, R. Ditchfield. M.D. Xe\e\iton and J-A. Popk, GAUSSIAN 70. Progam 236, Quantum Chemistry Progam Exchange. indmna University-

bf. Dupuis, J. R]s and K. F_ Ring, HONDO 76, Program 338, Quantum Chemistry Program Exchange, Indi-ana University.

Volume 65. number 1 CHEMICAL PHYSICS LETTERS 1 Auglst 1979

[221 S-W_ Benson. J_ Chem. Phys. 34 (1961) 521; E. O’NeaI and S-W_ Benson, J. Phys. Chem. 72 (1968) 1866.

[231 R_ Hoffman. J_ Am_ Chem. Sot_ 90 (1968) 14bS. [241 L_ Salem. in: The world of quantum chemistry, edr B. Pullman and RG. Parr (Reid& Dordrecht, 1976) p- 2441. [ZS] J.A. Horsiey, Y. Jean, C_ Moser, L._ Salem. R.M_ Stevensand J.S. Wright, J. Am Chew_ Sor 94 (1972) 279. 1261 PJ. Hay, WJ. Hunt and W.A_ Goddard III, J_ Am_ Chem Sot_ 94 (1972) 634. [271 K. Yawuchi, A_ Nishio, S. Yabusbits znd I_ Fueno. Chem Phqs_ Letters 53 (1978) 109. [2S] J-A. Berson and L_D. Pederscn, J_ Am. Chem. Sot. 97 (1975) 236.