Volume 125, number 3 CHEMICAL PHYSICS LETTERS 4 April 1986 ON THE DIRECT DETERMINATION OF CONSTRAINED PURE STATE ONE-ELECTRON DENSITY MATRICES: A NEW ALGORITHM Kalyan K. DAS and S.P. BHATTACHARYYA Theory Group, Department of Ph.vsical Chemrstry, Indtan Assocrairon for the Cultruatron of Sctence, Jadavpur. Calcutta 700 032, India Received 30 July 1985; in final form 25 January 1986 An algorithm based on a “constrained variational principle” is suggested for the direct determinatton of constramed one-electron density matrices in a self-consistent manner. The algorithm uses both penalty-function and Lagrangian multiplier methods of mcorporating equality constraints and can tackle any number of constraints. Results of preliminary calculations are presented. 1. Introduction The ground-state electronic structure of closed- shell atoms and molecules is often well represented by a single-determinant wavefunction constructed from a set of variational optimized orthonormal one- electron orbitals (the Hartree-Fock orbitals [l-4] ). Alternatively one may treat the elements of the one- electron density matrix (P) as the basis variables and determine P directly by invoking the variational prin- ciple [.5,6]. In either case, one essentially solves a constrained energy variational problem, the con- straint being the orthonormality of the HF orbitals. One may think of a more general type of constrained variational principle [7-91. The need for doing so can be rotationalized by noting that the variationally determined wavefunction is correct only to first order and therefore the expectation values of operators which do not commute with the Hamiltonian operator Q of the system, calculated from this variationally selected wavefunction, are correct only to first order, unlike the energy which is correct to second order. It is possible in this context to think of a more general constrained variational principle. We ask: Is it possible to determine a set of one-electron orbitals or the cor- responding one-electron density such that the energy as well as the mean-square deviation of a set of cal- culated values of observables from their experimental 0 009-26 14/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) or accurate theoretical counterparts is minimized? If the answer is in the affirmative, these “property- constrained” orbitals or one-electron density could be used to define an alternative zeroth-order wavefunction for perturbative or configuration interaction calcula- tions, presumably with faster “convergence” with respect to a number of properties, depending on the nature of the particular property (properties) used to constrain the density or the wavefunction. Even the convergence with respect to energy could perhaps be accelerated by an appropriate choice of the constraint. Mukherjee and Karplus [7] were the first to show ex- plicitly that by forcing the wavefunction to satisfy one additional “property constraint”, the calculated values of a number of observables could be improved simultaneously with a small sacrifice in the calculated energy [7]. However, the approach adopted for solving the multiply constrained variational problem was a brute-force one and cannot be used as a generally use- ful procedure. Rasiel and Whitman [9] elaborated the work of Mukherjee and Karplus and tried to formalise the entire procedure. Even then, this cannot be claimed to have developed into a generally practicable technique of solving the multiply constrained variational problem. A thorough theoretical analysis of the different as- pects of the problem was made by Byers Brown [8] and later by Chong [lo], revealing the possibility of making useful application of this kind of constrained 22.5