Analytical gradients orbital

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

Fournier, R., Andzelm, J., Salahub, D. R., 1989, Analytical Gradient of the Linear Combination of Gaussian-Type Orbitals-Local Spin Density Energy , J. Chem. Phys., 90, 6371. 

A promising development in the latter direction was the implementation of a multicon-figurational DFT approach [52] with empirical parameters (in addition to those already contained in the mixed density functionals). This method can yield accurate potential energy surfaces of excited states, and has recently been adapted to perform spin-orbit Cl calculations [53], but the lack of analytic gradients has up to now prevented its use in the simulation of molecular dynamics and in geometry optimizations. A more classically... 

To determine the virtual orbitals which minimise the variational energy (15), the derivatives with respect to the basis set expansion, see Eq. (13), and the configuration coefficients, Eq. (11) are computed. Analytic gradients and second derivatives, including mixed terms, are computed and inserted into the Newton-Raphson stabilised algorithm [19]. The detailed expression of the derivatives... 

Tel. 904-392-1597, fax. 904-392-8722, e-mail aces2 qtp.ufl.edu Ab initio molecular orbital code specializing in the evaluation of the correlation energy using many-body perturbation theory and coupled-cluster theory. Analytic gradients of the energy available at MBPT(2), MBPT(3), MBPT(4), and CC levels for restricted and unrestricted Hartree-Fock reference functions. MBPT(2) and CC gradients. Also available for ROHE reference functions. UNIX workstations. 

C. D. Sherrill, A. I. Krylov, E. F. C. Byrd, and M. Head-Gordon, /. Chem. Phys., 109, 4171 (1998). Energies and Analytic Gradients for a Coupled-Cluster Doubles Model Using Variational Brueckner Orbitals. Application to Symmetry Breaking in O4. 

For MBPT and CC methods, evaluation of the reduced density requires determining a response vector (A) as well as T. This defines a response density p = e Oo)(o (l + A)e. In addition, we want to allow the molecular orbitals to relax. The latter consideration adds another term, p", to the one-particle density. This relaxed density, p = p -I- p", is the critical quantity in CC and MBPT analytical gradient (and property) methods. " For just the one-particle part, we have p(l) = p (l) -I- p" = D(l) which will show up again when we discuss properties. 

To discuss the form and cost of analytic gradient and Hessian evaluations, we consider the simple case of Hartree-Fock (HF) calculations. In nearly all chemical applications of HF theory, the molecular orbitals (MOs) are represented by a linear combination of atomic orbitals (LCAO). In the context of most electronic structure methods, the LCAO approximation employs a more convenient set of basis functions such as contracted Gaussians, rather than using actual atomic orbitals. Taken together, the collection of basis functions used to represent the atomic orbitals comprises a basis set. 

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