As a demonstration of the accuracy and applicability of the proposed calculation algorithm, essentially exact potential BYL719 purchase energy curves of few-electron molecular systems with long interatomic distances are described for cases where the conventional calculation methods of quantum chemistry fail. The organization of the article is as follows. In the ‘Optimization algorithm’ section, Selleck AZD5153 the proposed calculation algorithm for constructing a basis set
of nonorthogonal SDs by updating one-electron wave functions with multiple correction vectors is described. The expression of the conventional steepest descent direction with a Gaussian basis set is also given for comparison. The convergence characteristics to the ground states of few-electron systems for calculations using single and multiple correction vectors are illustrated in the ‘Applications selleckchem to few-electron molecular
systems’ section. As demonstrations of the proposed calculation procedure, the convergence properties to the ground states of few-electron atomic and molecular systems are also shown. Finally, a summary of the present study is given in the ‘Conclusions’ section. Optimization algorithm The calculation procedures for constructing a basis set consisting of nonorthogonal SDs for N-electron systems using single and multiple correction vectors are described here. An N-electron wave function ψ(r 1, σ 1, r 2, σ 2,…, r N , σ N ) is expressed by a linear combination of nonorthogonal SDs as follows: (1) Here, r i and σ t denote the position and spin index of the ith electron, respectively. L is the number of SDs, and Φ A (r 1, σ 1, r 2, σ 2,…, r N , σ N ) is the Ath SD, given by (2)
(3) with ϕ i A (r) and γ i (σ i ) being nonorthogonal and unnormalized one-electron basis functions and spin orbital functions, respectively. The one-electron Florfenicol wave function ϕ i A (r) is constructed as a linear combination of Gaussian basis functions x s (r)  as (4) Here, M and D i,s A are the number of basis functions and the sth expansion coefficient for the ith one-electron wave function ϕ i A (r), respectively. The steepest direction is implemented in the expression of the total energy functional E of the target system on the basis of the variational principle, without the constraints of orthogonality and normalization on the one-electron wave functions. The updating procedure of the pth one-electron wave function belongs to the Ath SD which is represented as (5) where a p A is the acceleration parameter, which is determined by the variational principle with respect to the total energy E, i.e.,  (6) The component of the steepest descent vector K p,m A is given by (7) where (8) (9) and (10) Here, denotes the element of the jth row and ith column of the matrix .