Moitra, R K

Mukherjee, Debashis

Mukhopadhyay, A

English

IACS Institutional Repository

Indian J . P hys. 48, 472-474 (1974)On a note concerning the linked-cluster factorisation in an open-shell theoryD ebash is  M u k h b r jb eD ep artm en t o f  C h em istry , P resid en cy  College, Calcutta 700012,ANDR . K . M o itb a  an d  a . Mu k h o pa d h y a ySaha InstittU e o f  N u clea r  P h y s ic s , Calcutta 700009 {R eceived  17 A u g u st  1973)In this note we present a few formulae related to tho linked-oluster factorisation in a non-perturbatic approach to the calculation of Correlation energy of a de­generate or nearly degenerate system. A similar theory for the non-degenerate systems has earlier been given by Cizek (1966). With the starting wave-fiino- tion ^0 to be a combination of components (j>i of the form\ ^ o >  ... (1)ithe exact wave-ftmction | ^ e x  >  mfi^ y always be generated by the action of an excitation  operator e^ on... (2)We have found that the matrix-elements of the operators in S  may be deter­mined from the Schrodinger equation for | >  using various identities andoperator equalities. In this connection, we have derived a theorem which is ol central importance in providing the desirable property of a linked-cluster factori­sation of the relevant equations. The theorem states that, if, for the choice of any of the components as the vacuum the contractions of the operators in ^ are zero, then the following relation holds good=  e ^ U i \ ^ i > ,  ... (3)where H  is the Hamiltonian, and U{ is a set of linked contracted operators. Writing S  in the form... (4)f  8A.where S p  is the p-particle excita tion  operator for an N-eleotron system, the operator Ui is given by1Sk S n  fri, ifcfl ’« 'l r  « 2  ~  0 ,  k(S)mOn a note concerning the Unked-cluster 473where /  and v  are the one- and two-partiole parts of the Hamiltonian put in n orm al order with respect to as the vacuum. The subscript H' on {f-V'V) denotes that it is dependent on the choice of >  as tlie vacuum. The (piaatity {R^ ... R k i ... consists of a number of terms, all of which arelinked in the sense that the associated product of operators do not factor out as mutually uncontracted subsets.H e s  I s  CiUi\(l>i >  E e s  2  C ^ ui iUsing this theorem, the Schrodingcsr equation may bo writtcni in the form... (6a)tand the exact energy E  is given by... (6b)£  =  S C tC )  <  <P}\ui\<f>tt.jThe equations determining the niatxix-elemcmts of S  may bo found out by projecting eq. (6a) onto the excited states >  orthogonal to all and are given byfor all i and m .<  <f>m* 1 I >  = 0 ,Defining an operator u  by the relationU \ ^ t >  =  U i\ ^ ithe o q , (6b) may be re-written as followsJSJ =  2  C iC ^t(6c)(7)(8)The operator u may thus bo thought of as an effective oj)erator which acting entirely within the space of the compoiumts <pi generates the exact energy of the state \(j>ex >• I^om the manner of construction of it is clear that they are independent of the coefficients C i so that for k components 1 > ,  the exact energies E^  of the k states having starting wave-functions may be found out from the relation(9)For states where the coefficients are determined purely by symmetry, the energies JS?* are known whenever are known. However when are not determined by symmetry alone, as is the case for near degenerate systems, they may be found out from the condition that the associated energies E ^  are oxtremum. This leads to an eigen-value equation of the formla < ^ i \ u \ i > t >  C i^  ^  E ^ C j . (10)12By choosing the operators in S  to bo anti-hormitoan, the effective operator u  may always bo cliosen to be hormitean, so that solution of the eigen-value oq. (10) leads to real energies E  and orthonormal wave-functions Vre®*- Thus, the use of the theorem described in this note enables one to define an effective operator u , which may bo made hermitean, such that it generates the exact energy eigen-values of various states by acting entirely within the spacespanned by the components (j>i. The restrictions imposed on 8 are fairly trivial, so that the tlieorem holds true under quite general conditions.R e f e r e n c e sCizuk J. 1966 J. Chenu Phys, 45, 4256.474 D . Mukherjee, R . K . M itra and A . M ukhopadhyayIndian J , Phya. 48. 474-470 (1974)On the choice of virtual orbitals in a many-body perturbation calculation of pair-correlation energiesD e b a s h i s  M u k h e r j e eD ejm rtm en t o f  C h em istri/, P re s id e n c y  C ollege, C alcu tta  7 0 0 0 1 2 ,  R . K. M o ITRA a n d  a . M U K H O rA D H Y A Y  S a fm  In stitu te  o f  N u c lea r  P h y s ic s , C a lcu tta  7 0 0 0 0 9  (R eceived  17 A v g u s t  1973)In a perturbation calculation of the Correlation energy, one usually starts with the Hartree-Fock (HF) function as the unperturbed wave-function, and improves on the associated energy by a perturbation expansion. The partition of the total electronic Hamiltonian H  into an unperturbed part H q and a perturbation J/j in a perturbation calculation may be achieved in many ways. However, only those partitionings are preferred which keep the perturbation as small as possible. The oldest partitioning is that of Moller and Plosset (MP), whore is taken to bo a sum of HF one-electron Hamiltonian. This partition introduces large diagonal matrix-elements of the pertui’bation in the associated perturba­tion series and thus suffers from a slow convergence. A vastly bettor partition is, however, that of Epstein and Nosbot (EN) where H q is so chosen that the above matrix-elements do not appear at all. In this partition, the diagonal

On a Note Concerning the Linked-Cluster Factorisation in an Open-Shell Theory

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On a Note Concerning the Linked-Cluster Factorisation in an Open-Shell Theory

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