Gradients in valence bond theory

Abstract

A gradient method for general valence bond wave functions is presented. The electronic energy is used as a Lagrange multiplier. The derivatives of the normalization and of the first-and second-order cofactors present in the energy expression have to be evaluated, giving rise to first-, second-, and third-order cofactors. This evaluation is done using an extension of methods described previously. The use of gradients is illustrated with some calculations on organic molecules, viz. ethene, 1, 4-butadiene, and benzene.

1,3,5-Cyclohexatriene captured in computro; the importance of resonance

“…The geometries of benzene ðD 6h -2; W 2 Þ and 1,3,5-cyclohexatriene ðD 3h -1; W 1 Þ were optimized using gradient techniques [15]; a 6-31G basis set [16] was used. The calculations were performed using the Ab Initio Valence Bond program TUR-TLE AMESS-UK TLE [17], as integrated in the GAMESS-UK package [18].…”

Reply to comment on the paper “An efficient Algorithm for Energy Gradients and Orbital Optimization in Valence Bond Theory”

“…However, our algorithm is designed in such a way that the Fock matrix, constructed from the transition density matrix between two determinants, can be used to evaluate the firstorder energy derivative with respect to the orbital coefficients (for more details, see Eq. (31) in our original paper [3] ). This is at the heart of our whole algorithm which consequently scales at O(m 4 ) for one determinant pair.…”

“…While we stand by our statement that our algorithms scales at O(m The Comment by van Lenthe, Broer, and Rashid [J. H. van Lenthe, R. Broer, Z. Rashid, submitted] criticized the title paper [1] on the grounds that (1) the paper neglects to properly reference the much earlier work by Broer and Nieuwpoort, [2] who developed an algorithm to use a Fock matrix to compute a matrix element between two different determinants, and (2) the paper contains a misleading comparison with van Lenthe et al's valence bond self-consistent field (VBSCF) algorithm. [3] In addition, these authors made a few more remarks on our original paper. First of all, the Broer-Nieuwpoort work is ' 'concentrated on a broken symmetry approach to the description of certain excitations and ionizations in systems with spatial symmetry,' ' and their algorithm for the evaluations of matrix elements between determinants of nonorthogonal orbitals, presented in the Appendix, is based on the biorthogonal transformations.…”

On the efficiency of VBSCF algorithms, a comment on “An efficient algorithm for energy gradients and orbital optimization in valence bond theory”

“…Recently, a scheme was published, where just the gradient was calculated for a few orbital mixings using transition density matrices. [8] To make a comparison with the time needed to calculate the Lagrangian [9] we made a very rough guess, which we will reproduce here. The number of singly excited states (Brillouin states), which depends on the number of basis functions n and the number of active orbitals N is approximately nN.…”

“…In a complete Super-CI calculation, we need the interaction between all the Brillouin states hW kl |HÀE 0 |W ij i, a total of (nN) 2 matrix elements. As one matrix element requires $N 4 operations, we estimated a total number of operations of n 2 N 6 , [9] ignoring the triangle symmetry (a factor of 2). The Lagrangian, [10] at a cost of $N 6 , does not feature in the To get a more precise estimate of the number of Brillouin states, we have to consider the different types of orbitals featuring in the singly excited states.…”