Hartree-Fock second derivatives

Wong MW, Wiberg KB, Frisch M (1991b) Hartree-Fock second derivatives and electric field properties in a solvent reaction field Theory and application. J Chem Phys 95 8991-8998... 

M. W. Wong, K. B. Wiberg, and M. j. Frisch,/. Chem. Phys., 95, 8991 (1991). Hartree-Fock Second Derivatives and Electric Field Properties in a Solvent Reaction Field Theory and Application. 

Molecular frequencies depend on the second derivative of the energy with respect to the nuclear positions. Analytic second derivatives are available for the Hartree-Fock (HF keyword). Density Functional Theory (primarily the B3LYP keyword in this book), second-order Moller-Plesset (MP2 keyword) and CASSCF (CASSCF keyword) theoretical procedures. Numeric second derivatives—which are much more time consuming—are available for other methods. 

Gaussian can also predict some other properties dependent on the second and h er derivatives of the energy, such as the polarizabilities and hyperpolarizabilities. These depend on the second derivative with respect to an electric field, and are included automatically in every Hartree-Fock frequency calculation. 

The fii st term is zero because I and its derivatives are orthogonal. The fourth term involves second moments and we use the coupled Hartree-Fock procedure to find the terms requiring the first derivative of the wavefunction. 

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). 

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 SCF-MI algorithm, recently extended to compute analytic gradients and second derivatives [18,41], furnishes the Hartree Fock wavefunction for the interacting molecules and also provides automatic geometry optimisation and vibrational analysis in the harmonic approximation for the supersystems. The Ml strategy has been implemented into GAMESS-US package [42]. 

Further we can proceed similarly as in the case of adiabatic approximation. We shall not present here the details, these are presented in [21,22]. We just mention the most important features of our transformation (46-50). Firstly, when passing from crude adiabatic to adiabatic approximation the force constant changed from second derivative of electron-nuclei interaction ufcF to second derivative of Hartree-Fock energy Therefore when performing transformation (46-50) we expect change offeree constant and therefore change of the vibrational part of Hamiltonian... 

Hartree-Fock models are well defined and yield unique properties. They are both size consistent and variational. Not only may energies and wavefunctions be evaluated from purely analytical (as opposed to numerical) methods, but so too may first and second energy derivatives. This makes such important tasks as geometry optimization (which requires first derivatives) and determination of vibrational frequencies (which requires second derivatives) routine. Hartree-Fock models and are presently applicable to molecules comprising upwards of 50 to 100 atoms. 

Despite the fact that numerical integration is involved, pseudoanalytical procedures have been developed for calculation of first and second energy derivatives. This means that density functional models, like Hartree-Fock models are routinely applicable to determination of equilibrium and transition-state geometries and of vibrational frequencies. 

For quantum chemistry, first-row transition metal complexes are perhaps the most difficult systems to treat. First, complex open-shell states and spin couplings are much more difficult to deal with than closed-shell main group compounds. Second, the Hartree—Fock method, which underlies all accurate treatments in wavefunction-based theories, is a very poor starting point and is plagued by multiple instabilities that all represent different chemical resonance structures. On the other hand, density functional theory (DFT) often provides reasonably good structures and energies at an affordable computational cost. Properties, in particular magnetic properties, derived from DFT are often of somewhat more limited accuracy but are still useful for the interpretation of experimental data. 

The second chapter introduces the student to orbitals proper and offers a simplified rationalization for why orbital interaction theory may be expected to work. It does so by means of a qualitative discussion of Hartree-Fock theory. A detailed derivation of Hartree-Fock theory making only the simplifying concession that all wave functions are real is provided in Appendix A. Some connection is made to the results of ab initio quantum chemical calculations. Postgraduate students can benefit from carrying out a project based on such calculations on a system related to their own research interests. A few exercises are provided to direct the student. For the purpose of undergraduate instruction, this chapter and Appendix A may be skipped, and the essential arguments and conclusions are provided to the students in a single lecture as the introduction to Chapter 3. 

Derive the detailed expression for the orbital Hessian for the special case of a closed shell single determinant wave function. Compare with equation (4 53) to check the result. The equation can be used to construct a second order optimization scheme in Hartree-Fock theory. What are the advantages and disadvantages of such a scheme compared to the conventional first order methods ... 

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