Volume 142, number 6 CHEMICAL PHYSICS LETTERS 25 December 1987

ANALYTICAL ENERGY GRADIENTS FOR A UNITARY COUPLED-CLUSTER THEORY

Mark R. HOFFMANN and Jack SIMONSDepartment o[Chemistry, University o[Utah, Salt Lake City, UT 84 JJ2, USA

Received 9 September 1987; in final form 16 October 1987

Our recently developed and tested unitary multiconfigurationaI coupled-cluster electronic wavefunction method is extended to

permit, for the first time, the analytical evaluation of energy derivatives. The unitary nature of this method admits a variational

energy functional whose stationary nature plays a key role in sirnplifying our derivation. Explicit expressions are given for the

gradient (first energy derivative) for both the f~l1 unitary coupled cluster and its coupled e1ectron pair approxirnation ( CEPA).

.Recently, we [I] developed a the~ry of a unitary coupled-cluster (UCC) method and demonstrated its com-putational applicability via illustrative calculations on the well-studied [2-6] BeH2and H2O species. The un-itary nature of our method, and the resultant variational energy expression, distinguish our approach from thoseof most others and allow us to straightforwardly obtain a compact analytical expression for the UCC energygradient.

The importance of analytic derivativesin investigating chemical phenomena on a Bom-Oppenheimer po-tential-energy hypersurface is widely appreciated [7,8]. In this Letter, we demonstrate that a class of energyfunctionals, of which our UCC energy is a member, can be readily differentiated using procedures similal' tothose used in evaluating configuration interaction (CI) [9,10] and multiconfiguration self-consistent field (MCSCF) [10-14] energy derivatives. We provide explicit formulas for the UCC gradient in terms of integrals anddensity matrices over moleculal' and atomie orbitais as well as orbital and configuration amplitude responseswhich are obtained through coupled perturbed MC SCF methods [11-14]. Energy-derivative exprt:ssions foran important approximation to the coupled-cluster wavefunction, the coupled electron pair appróximation(CEPA) [15], are also included.

The variational condition that the unitary coupled-cluster energy (ref. [I], eq. (2.1I» is stationary withrespect to variations in the cluster amplitudes (óEcclótK=O) was shown in ref. [I] to lead to the followingexpression for the UCC energy:

Ecc= «PMc lexp( -T) Hexp(T) I<PMc)=EMC+ L«PMc IHIK)tK'K(I)

which is valid through second order in the t K amplitudes. Here the summation is over the configuration statefunctions (CSFs) in the space which interacts with the MC SCF reference function I<PMc)through the Ham-iltonian H, and t K is an amplitude of our UCC cluster operator T coupling I<PMc)to IK): t K= <KI TI<PMc>.In eq. (I), EMc is the energy of the MC SCF reference state.

In the CEPA version [16] of our UCC method [1], the t Kare calculated directly as variational parameters;in the UCC method, the t Kare obtained as <KI TI <PMc> after the variational parameters t", (which defineT= L",t",e",in terms of generators eijand single products of such gener~tors of the unitary group) are computedbysolvingthe linearequationsofref. [I]. Boththe CEPAand UCCm~thodsgiveidenticalenergyexpressions.

The conventional one. and two-electron operators forming the Hamiltonian may be introduced to give

EcC =EMc + 'f.hijDij + 'f. (ijl kl)dijkl ,ij ijkl

(2)

0009-2614/87/$ 03.50 @ Elsevier Science Publishers B.V.~(North-Holland Physics Publishing Division)

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Volume 142. number 6 CHEMICAL PHYSICS LETTERS 2S December 1987

a form reminiscent of the CI or MC SCF energy expressions, although here the density matrices dijkland Dijare of the transition density matrix form. Explicitly

Dij=LTKLCI(lleijIK) ,K I

(3a)

,and

dijkl=iLTKLC/( (Ileijk/IK) + (IIeijkIIK» ,K I

(3b)

where CI is the CSF amplitude in the MC SCF reference function and eijand eijklare generators and generatorproducts of the unitary group. The geometry dependence of Ecc is contained in the t K,the CI and the integralshij and Wlk/).

Application of the chain rule for differentiation and using the fact that the energy has been made stationarywith respect to variations in the TKor t" cluster amplitudes (aECEPA/aTK=Oor aEcc/at" = O) aUowsone to write

dEcc dEMc "huQD "ha D +"h aDij" (""

lk/) UQ d + "d (I )a +" (""

lk/)adijk'-

d=-

d + I... ij ij + I... iJP iJP I... ij :> + I... l) ijk' I... iJPP<1jlV pa I... lJ :>'a a ij iJP ij ua i;kl iJP ijkl ua

(4)

The variational nature of Ecc with respect to the TK(or t,,) amplitudes obviates the need to solve for the re-sponse oj the CC amplitudes with respect to geometrical displacement. Only the orbital responses Uj and theMC SCF configuration amplitude responses ac/aa need to be obtained [I 1,12]. The practical evaluation 9four unitary CC energy derivative (eq. (4» requires little more than the effort required for the MC SCF gra-dient and Uj and ac,/aa evaluation. This is a very important practical feature of our result.

In eq. (4), the hHQ and Wlk/ )uQare partiallytransformed[10,13,14]one-and two-electronintegralsin whichthe orbital responses l.jU'j;f/Jjhave been included as described in ref. [13], DiJPand diJPpqare density matriceswhich have been back-transformed to theatomic orbital basis [9]. For unitary CEPA energies, for which theTK themselves are the independent variational parameters, the transition density matrix derivatives aDi;laaandddikl/aaare expressed as in eq. (3) but with ac,/aa in place of CI, Since the TK in a UCC calculation are notthe independent variational parameters (the t" are), the derivative density matrices contain both this ac,/oafactor plus a second term ofthe same form as in eq. (3) but with atK/aa=l.L(KI TIL)aCdaa in place ofrK.With these specifications, eq. (4) gives or working resuit for the UCC and variational CEPA first energyderivative.

We again stress that aU quantities in eq. (4) are available either from the MC SCF gradient calculation, theUCC energy calculation, or by performing the coupled perturbed multiconfiguration Hartree-Fock (CP MCHF) response calculation [10- 12] to compute the molecular orbital and CI response quantities (ac,/aa andUj). The usual form ofthe CP MC HF equations [10-13] (Le. as used in MC SCF second derivatives andin CI first derivatives from an MC reference) may be used straightforwardly whenever the electronic energyexpression is invariant to unitary transformations within the core valence, and virtual molecular orbital sub-sets. This is, in fact, the case in the CEPA implementation of our DCC method. .

The implementation of our UCC method as expressed in ref. [l] may, for practical reasons, utilize a (nu-mericaUy) screened manifold of cluster operators. As a result, the UCC energy is, in general, not invariant toarbitrary orbital rotations because this screening process will usuaUydestroy invariance within orbital subsets.Therefore, a specific choice of the orbitals (e.g., some physicaUy motivated choice of canonical orbitaIs) mustbe made and preserved at aUnuclear geometries in order to have a consistently defined wavefunction. As de-scribed in our earlier paper, the specific choice of orbitaIs that we make diagonaliies the core Fock matrix[Il,17], ,

occ

Fij =hij+l L Ak/[2WI ki) - (iklj/)],ki(5)

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Volume 142, number 6 CHEMICAL PHYSICS LETIERS 25 December 1987

within both the core and virtual orbital spaees, and diagonalizes the Lagrangian matrix [11,18],

E;j=rh;ki'Ig+2 > (ikl/m)rjklm ,k k'rm

(6)

in the valence orbital space.The eondition that the moleeular orbitals at the (infinitesimally) displaeed geometries also diagonalize

Fij and Eijand henee that the eoupled-cluster operator manifold at the slightly displaeed geometry is consistentwith the manifold at the original geometry, gives rise to eonditions on the derivatives of the eore Fock matm[11 l,

dFijlda=O, it-j

and on the derivatives of the sum of the off-diagonal Lagrangian matrix,

d(E;j+Ej;)/da=O, it-j. (8)

The difference of the off-diagonal Lagrangian matrix elements must also vanish (at alI powers of differentia-tion), a condition already used in defining the CP MC HF equations [11 l.

Differentiating the core Foek matrix, as indieated in eq. (7), gives

(7)

- 0= rC~IY. (h~ + ri'pa[(Jllllpq)Q - l(ppl VO')Q] )+r(r 0;1 CAi'ft+i'tf»)[(ijlkl) - l(iklj/)]pJl pa ki IJ ua

+ ~u~{ ~i'd(sjlk/) - Hsklj/)]+hsj)+ ~u~( ~i'kl[(islkl)- l(ikls/)] +h;$)

+ f,u~,( ~i',,[2WlsI) -Hislj/) - l(ilIjS)])(9)

for all i>j, i,jeeore and i,jevirtual. In eq. (9), the Greek indices run over all atomie orbitals, k, I, r denotemoleeular orbitals occupied in some or all CSFs, and s runs over all moleeular orbitals. The terms in eq. (9)are identical to those derived earlier by Osamura and eo-workers [II] in the eontext of derivatives of a CIwavefunetion osing MC SCF orbitals. However, in the present case, the separation into energy redundant andnon-redundant variabIes does not oeeur. Thus, eq. (9) represents a generalization of the coupled perturbedmultieonfigurational Hartree-Fock (CP MC HF) equations ofOsamura and co-workers for the UZand oC1loa.

Similarly, the sum of the off-diagonal Lagrangian matm elements ean be differentiated to give

O=rU~;E$j+rU~'Y;j$,+Ez+r 0;1 n+(i+=tj),,$ $' 1 ua

(10)

for all i>j, i,je valenee. In eq. (10), we use the same index convention deseribed in the preeeding paragraph.Examination of eqs. (9) and (10) shows that the usual energy non-redundant CP MC HF equations must

be modified by adding nco",(""o",+ 1)/2 + nvalence(nV81ence+ 1)/2 + nvinU81(nVinU81+ 1)/2 equations to preserve thecanonical nature of the orbitais. Further examination of eqs. (9) and (10), with consideration of the orbitalrotation orthonormality condition, UZ+ Uj;+ SZ =O (where SZ is the usual derivative overlap matrix trans-formed into the MO basis), shows that the number oforbital rotations (UZ) introdueed in excess ofthe energynon-redundant ones exaetly equals the number of additional equations. Thus it is to be expeeted tbat the larger.set of simultaneous equations has a unique well-defmed solution. Parenthetically we note that the quantitiesused to eonstruet the eoeffieients in the additional equations and t~ modifications to existing CP MC HFequations are already available within conventional computer codes for the construction ofthe usual CP MC HFequations [11,12]. Tbe solutions to these modified equations are then the orbital (UZ), and configuration am-plitude (oCrloa) changes whieh may then be used directly in computing the derivative energy, as in eq. (4).

,~-~-,. "-" \ ;~'-?A "': ..

;,..,/J",!",'\' ;, -,I ,-' ,--y 'ji::'" , :I ",' -'. t ,..-i'; '.;"'" _. ~" "" ,', r - ' '- -

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Volume 142, number 6 CHEMICAL PHYSICS LETTERS 25 December 1987

In summary, we have demonstrated in this Letter that the UCC molecular gradient, or its variational CEPAcounterpart, are indeed practical within the framework of our UCC method. The unitary method and its re-suIting variational energy functional are the key ingredients to obtaining a computationaIly practical energygradient expression. ComputationaI implementation of this method is eminently viable due to both thestraightforward functional form oftIt~ energy gradient expression and to the availability ofkey quantities fromextant MC SCF analytical derivative codes.

We acknowledgethe financial support of the NationaI Science Foundation (CHE-8511307) and the US ArmyResearch.Office(DAAG-2984KOO86),and the donors ofthe Petroleum Research Office Grant (PRF 14446AC6)administered by the American Chemical Society.

References

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[3] A. Banerjee and J. Simons, Chem. Phys. 81 (1983) 297; 87 (1984) 215.[4] M.R. HofTmanri,X.-c. Wang and K.F. Freed, Chero. Phys. Letters 136 (1987) 392.[5] P. Saxe, H.F. Schaefer III and N.C. Handy, Chem. Phys. Letters 79 (1981 ) 202;

RJ. Harrison and N.C. Handy, Chem. Phys. Letters 95 (1983) 386.[6] RJ. Bart1ett, H. Sekino and G.D. Purvis, Chem. Phys. Letters 98 (1983) 66;

W.D. Laidig and R.J. Bart1ett, Chem. Phys. Letters 104 (J 984) 424.[7] H.F. Schaefer III, Quantum chemistry (Clarendon Press, Oxford, 1984).[8] P. J0rgensen and J. Simons, eds., Geometrical derivatives of energy surfaces and molecular propertie$'( ReideI, Dordrecht, 1986).[9] B.R. Brooks, W.D. Laidig, P. Saxe, N.C. Handy and H.F. Schaefer III, Physica Scripta 21 (1980) 312;

B.R. Brooks, W.D. Laidig, P. Saxe, J.D. Goddard, Y. Yamaguchi and H.F. Schaefer III, J. Chem. Phys. 72 (J980) 4652.[10] P. J0rgensen and J. Simons, J. Chem. Phys. 79 (1983) 334.[11] Y. Osamura, Y. Yamaguchi and H.F. Schaefer III, J. Chem. Phys. 77 (J 982) 383.[12] M.R. HofTmann, D.J. Fox, J.F. Gaw, Y. Osamura, Y. Yamaguchi, R.S. Grev, G. Fitzgerald, H.F. Schaefer III, P.J. Knowles and

N.C. Handy, J. Chem. Phys. 80 (1984) 2660.[13] M. Page, P. Saxe, G.F. Adams and B.H. Lengsfield, J. Chem. Phys. 81 (1984) 434.[14] T.U. Helgaker and J. Almlof, Intern. J. Quantum Chem. 26 (1984) 275.[15] W. Kutzelnigg, in: Methods ofelectronic structure theory, Vol. 3, ed. H.F. Schaefer III (Plenum Press, New York, 1977) pp. 129-188,

and references therein.

[ 16] M.R. HofTmann and J. Simons, to be submitted for publication.[17] P. Siegbahn, A. Heilberg, B. Roos and B. Levy, Physica Scripta 21 (1980) 323.[18] K. Reudenberg, L.M. Cheung and S.T. Elbert, Intern. J. Quantum Chem. 16 (1979) 1069.

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