CPHF coefficients

The matrix W is the so-called CPHF coefficients at the given geometry x and satisfies... 

Equation (2.19) may be further transformed to avoid the explicit evaluation of the derivative of the MO coefficients U°j with respect to the specific perturbation a. Using the orthonormality constraint of the perturbed orbitals, we can set the occupied-occupied, virtual-virtual and occupied-virtual CPHF coefficients to U j =... 

The block of vacant-occupied CPHF coefficients f/ of Eq. (2.20), which depend on the specific perturbation a, may be eliminated by using the interchange (Z-vector) method of Handy and Schaefer [20], properly extended to the PCM framework [21] ... 

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 CPHF procedure may be generalized to higher order. Extending the expansion to second-order allows derivation of an equation for the second-order change in the MO coefficients, by solving a second-order CPHF equation etc. 

MO coefficients. CPHF equations involve (first) derivatives of the one- and two-electron integrals with ... 

What has been accomplished The original expression (10.23) contains the derivative of the MO coefficients with respect to the perturbation (dc/dX), which can be obtained by solving the CPHF equations (Section 10.5). For geometry derivatives, for example, there will be 3N different perturbations, i.e. we need to solve 3N sets of CPHF... 

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. 

To be able to evaluate equation (43) we need the derivatives (bpfy/by) and (bW v/by), respectively. For this purpose we have to know the derivatives of the coefficients occurring in the COs, ( ( )/ ). They can be obtained from the solution of the coupled HF(CPHF) equations.49 Following the notation of Pople et al.,50 one can write... 

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 ... 

These equations can be derived by differentiating the expression for Emp2 and using the Z-vector method to avoid solving the CPHF equations explicitly for each of the derivatives of the MO coefficients. 

This is the expression based on real coordinates Ria. But the analogical expression based on normal ones Qr will be much more interesting. Therefore we introduce, again in accordance with the CPHF formulation, the expansion coefficients CpQ. 

Now we can proceed from the adiabatic limit to the B-O limit. In both approximations the same Eqs. 28.49 and 28.50 hold. The only but remarkable difference is the classical concept of the separation of degrees of freedom in the latter one. It means that the coefficients r, s in these equations represent only the normal vibrational modes. And besides in this simplified form the Eqs. 28.49 and 28.50 are exactly identical with the standard Pople s CPHF equations [23,24] after the formal rewrite from the fixed basis of atomic orbitals into the moving one, following the motion of nuclei. Since this is only a numerical problem, which does not affect the core of this topic, we only refer to preceding works [12,17]. 

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