Derivatives coupled perturbed Hartree-Fock

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. 

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. 

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. 

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. ... 

The fact that the gradient of the variational Cl energy does not contain the derivatives of the Cl coefficients has been pointed out early (Kumanova, 1972 Tachibana et al, 1978) indeed, this is implicit in some early work. However, no computationally attractive algorithm was given, in particular for the solution of the coupled perturbed Hartree-Fock (CPHF) equations, which are required for Cl gradients. 

On the contrary, the density derivatives are necessary to get higher order free energy derivatives in particular the second free energy derivatives require the first derivative of the density matrix P , which can be obtained by solving an appropriate coupled perturbed Hartree-Fock (CPHF) or Kohn-Sham (CPKS) equations. 

Once again, the derivative of the density matrix P can be obtained as solution of first-order coupled-perturbed Hartree-Fock equation with derivar tive Fock matrix given by eq. (1.70), exactly as for the nuclear shielding. 

The last three terms of this expression involve the derivatives of the molecular orbital coefficients and cannot easily be avoided. They are obtained through coupled perturbed Hartree-Fock theory (CPHF). ... 

The derivatives of the MO coefficients or the density matrix can be obtained by solving the the corresponding derivatives of the Hartree-Fock equations, i.e. the coupled perturbed Hartree-Fock (CPHF) equations. These can be solved either in the MO basis or in the AO basis. After some manipulation, the CPHF equations in the MO basis can be reduced to ... 

Wavefunction derivatives involved in Eq. [12] and [17] comprise the major part of the evaluations of AATs. In earlier days, a finite difference method was used to obtain these derivatives. The implementation of analytical coupled perturbed Hartree-Fock (CPHF) methods 3- 6 has permitted these derivatives to be obtained more efficiently. In the MFP approach, two sets of CPHF calculations are required to obtain re-... 

Unlike the true propagator, the UCHF approximation is given by a simple closed formula and reqnires only minimum computational effort to evalnate on the fly if the orbitals are available. The nnconpled Hartree-Fock/Kohn-Sham approximation has almost completely vanished from the chemistry literature about 40 years ago when modem derivative techniques became available because of the poor results it produced for second-order properties. Some systematic expositions of analytical derivative methods still use it as a starting point, but it is in our opinion pedagogi-cally inappropriate, as it requires considerable effort to recover the coupled-perturbed Hartree-Fock results which can be derived in a simpler way. UCHF/UCKS is still used in some approximate theories, but we suspect that its only merit is easy computability. According to Geerlings et al. [29], the polarizabilities derived from the uncoupled density response function correlate well with accurate results but can be off by up to a factor of 2, and thus they are only qualitatively useful. Our results in Table 1 confirm this. 

Factorization of the last term on the r.h.s. of Eq. (8) is harder, but it is facilitated by availabihty of quantum chan-ical software. From the coupled perturbed Hartree-Fock part of quantum chemical programs, it is possible to extract the first-order transformation matrix p9> which is needed for evaluation of derivatives of density matrix elements. 

In the previous equation R is the density matrix derivative with respect to Qi. It can be calculated throu a coupled perturbed Hartree-Fock (CPHF) procedure. Notice that the presence of the term, that is, of the dipole moment matrix derivative in Eq. 7.14, is due... 

P are calculated analytically via electric field derivatives of the total energy within a coupled perturbed Hartree-Fock approach. 

ASC = apparent surface charge BEM = boundary-element method CPHF = coupled perturbed Hartree-Fock C/RF = classical reaction field GBA = generalized Bom approximation FDM = finite-difference method FEM = finite-element method MPE = multipole expansion PD = potential derived SOS = sum over states SPT = scaled particle theory. 

The analytic determination of dipole moment derivatives by ab initio methods requires derivation of a complete formulation of coupled-perturbed Hartree-Fock equations by nuclear coordinate and electric field variations. 

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