Coupled perturbed Hartree Fock

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.29) gives directly an equation for determining the (first-order) response, which is structurally the same as eq. (10.36). For an HF wave function, an equation of the change in the MO coefficients may also be formulated from the Hartree-Fock equation, eq. (3.50). 

The superscript (0) here denotes the unperturbed system, molecular orbitals (eq. (3.20)) can be expressed as 

Expanding each of the F, C, S and e matrices in terms of a perturbation parameter (e.g. F= + AF + A F +. ..) and collecting all the first-order terms (analogous to the strategy used in Section 4.8) gives 

Equation (10.42) are the first-order Coupled Perturbed Hartree-Fock (CPHF) equations. The perturbed MO coefficients are given in terms of unperturbed quantities and the first-order Fock, Lagrange (a) and overlap matrices. The F term is given as (eq. (3.52)). 

The derivatives over the integrals may involve derivatives of the basis functions or the operator, or both (see Section 10.8). Using eqs. (10.45) and (10.44) in eq. (10.42) gives a set of linear equations relating to and 

Expan ng each of the F, C, S and e matrices in terms of a perturbation parameter (e.g, 

Note that no new operators are involved, only derivatives of the Cl or HF wave function with respect to the MO coefficients. The matrix elements can thus be calculated from the same integrals as the energy itself as discussed in Sections 3.3 and 4.2.1. The derivative with respect to a perturbation can now be written as in eq. (10.40). 

The second term disappears since the Cl wave function is variational in the state coefficients, eq. (10.36). The three terms involving the derivative of the MO coefficients (3c/32) also disappear owing to our choice of the Lagrange multipliers, eq. (10.39). If we furthermore adapt the definition that 3H/32 = Pi (eq. (10.20)), the final derivative may be written as eq. (10.42). 

The Lagrange technique may be generalized to other types of non-variational wave functions (MP and CC), and to higher order derivatives. It is found that the 2n -i-1 rule is recovered, i.e. if the wave function response is known to order n, the (2n -i- l)th-order property may be calculated for any type of wave function. 

Although a calculation of the wave function response can be avoided for the first derivative, it is necessary for second (and higher) derivatives. Eq. (10.32) gives directly an 

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CPHF (coupled perturbed Hartree-Fock) ah initio method used for computing nonlinear optical properties... 

Dispersion of Linear and Nonlinear Optical Properties of Benzene An Ab Initio Time-Dependent Coupled-Perturbed Hartree-Fock Study Shashi P. Kama, Gautam B. Talapatra and Paras N. Prasad Journal of Chemical Physics 95 (1991) 5873-5881... 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

As our computations use the HONDO/8 program (22) which is based on the CPHF (Coupled Perturbed Hartree Fock) method (23) we begin by briefly recalling this method. 

Here in eq. (38) "EpqfpQN a.pag is new Hartree-Fock operator for a new fermions (25), (26), operator Y,pQRsy>pQR a Oq 0s%] is a new fermion correlation operator and Escf is a new fermion Hartree-Fock energy. Our new basis set is obtained by diagonalizing the operator / from eq. (36). The new Fermi vacuum is renormalized Fermi vacuum and new fermions are renormalized electrons. The diagonalization of/ operator (36) leads Jo coupled perturbed Hartree-Fock (CPHF) equations [ 18-20]. Similarly operators br bt) corresponds to renormalized phonons. Using the quasiparticle canonical transformations (25-28) and the Wick theorem the V-E Hamiltonian takes the form... 

In many calculations beyond the Hartree-Fock level a first step is the transformation of at least some integrals. For the simplest such calculation, second-order perturbation theory, integrals with two indices transformed into the occupied MO basis axe required. Such integrals appear in many situations, including the MO basis formulation of coupled-perturbed Hartree-Fock theory. We can represent the first phase of this transformation as obtaining Coulomb and exchange operators ... 

The static polarizabilities, a, of various xanthone analogues including seleno- and telluroxanthen-9-one Id and le and seleno- and telluroxanthen-9-thione 2d and 2e have been estimated by ab initio molecular orbital calculations using the coupled perturbed Hartree-Fock (CPHF) method <1996CPL125>. The results indicate that the introduction of heavy elements in 1 and 2 increases all components of a with a greater effect observed in the case of the thione derivatives 2. 

There exist two main methods for implementing nonlinear optical calculations into a given computational technique coupled methods and uncoupled methods. Coupled methods [sometimes called finite field (FF) or coupled-perturbed Hartree-Fock (CPHF) methods] include the effect of the perturbing field into the Hamiltonian. The energy (e) of the system in the field E... 

The first derivative of the density matrix with respect to the magnetic induction (dPfiv/dBi) is obtained by solving the coupled-perturbed Hartree-Fock (or Kohn-Sham) equations to which the first derivative of the effective Fock (or Kohn-Sham) operator with respect to the magnetic induction contributes. Due to the use of GIAOs, specific corrections arising from the effective operator Hcnv describing the environment effects will appear. We refer to Ref. [28] for the PCM model and to Ref. [29] for the DPM within either a HF or DFT description of the solute molecule. 

Exact second derivatives methods require the solution of the coupled perturbed Hartree-Fock equations, CPHF [11,34,35]. At the Hartree-Fock level this requires several steps in addition to the usual SCF procedure and the evaluation of the first derivatives. 

We have also examined here the use of approximate solutions of the coupled perturbed Hartree-Fock equations for estimating the Hessian matrix. This Hessian appears to be more accurate than any updated Hessian we have been able to generate during the normal course of an optimization (usually the structure has optimized to within the specified tolerance before the Hessian is very accurate). For semi-empirical methods the use of this approximation in a Newton-like algorithm for minima appears optimal as demonstrated in Table 17. In ab-initio methods searching for minima, the BFGS procedure we describe is the best compromise. 

Coupled perturbed Hartree-Fock calculations at the SCF level were used to assess the polarizability tensor elements, each of which is defined as... 

Derivative Techniques 240 10.4 Lagrangian Techniques 242 10.5 Coupled Perturbed Hartree-Fock 244 10.6 Electric Field Perturbation 247 10.7 Magnetic Field Perturbation 248 10.7.1 External Magnetic Field 248 13.1 Vibrational Normal Coordinates 312 13.2 Energy of a Slater Determinant 314 13.3 Energy of a Cl Wave Function 315 Reference 315 14 Optimization Techniques 316... 

This method is valid only in the static field limit (zero frequency), which is a weakness. However, recent advances of a derived procedure (Coupled Perturbed Hartree-Fock) permit the frequency dependence of hyperpolarizabilities to be computed. The FF method mainly uses MNDO (modified neglect of diatomic differential overlap) semi-empirical algorithm and the associated parametrizations of AM-1 and PM-3, which are readily available in the popular MOPAC software package. ... 

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