SCF derivatives

Attempts to improve molecular wavefunctions so as to be able to calculate properties more accurately continue to be made, particularly via the constrained variational procedure. Two-particle hypervirial constraints were considered by Bjoma within the SCF formation,282 and he presented a perturbational approach to their solution.233 Using Scherr s wavefunction, and constraining p to satisfy the molecular virial theorem, a calculation on N2 led to rapid convergence.234-235 The constrained SCF orbitals are believed to be a closer approximation to the true tfi nearer the nucleus than further out. A later paper discussed the electron-density maps in comparison to the SCF derived maps, which confirm the conclusion that the wavefunction near the nucleus is improved.236... 

After omitting all contributions to the MCSCF derivatives which arise from variations in the configuration space, the remaining terms either already appear in the AO basis (e.g., all terms index transformed integrals.) This implies that SCF derivatives can be calculated completely in the AO basis. We give some examples to make this point clear. 

Pulay demonstrated that analytic first derivatives with respect to geometric parameters can be calculated easily and efficiently for HF energies. Derivatives of correlated methods followed a number of years after SCF derivatives [4, 5]. Extensions of the SCF derivatives to density functional theory methods were straightforward. In the three decades since Pulay s article, hundreds of papers on energy derivatives have been published, and all can trace their roots back to his paper. Energy derivatives have become so useful for calculating molecular structures and properties that, almost universally, first derivatives are formulated and coded soon after a new theoretical method is developed for the energy. 

All of the development of MC-SCF derivatives was performed in collaboration with H.B. Schlegel (Detroit). These codes have been interfaced with the GAUSSIAN system of programs. The applications work has been carried out in collaboration with... 

To overcome some of the problems inlierent in the UFIF method, it is possible to derive SCF equations based on minimizing the energy of a wavefiinction fomied by spin projecting a single Slater detemiinant starting... 

The gradient and second derivative components of the SCF energy can be expressed for both kinds of parametrization (see [28]) as... 

Drukker, K., Hammes-Schiffer, S. An analytical derivation of MC-SCF vibrational wave functions for the quantum dynamical simulation of multiple proton transfer reactions Initial application to protonated water chains. J. Chem. Phys. 107 (1997) 363-374. 

Usually, contractions are determined from atomic SCFcalculations. In these calculations one uses a relatively large basis of uncontracted Gaussians, optimizes all exponents, and determines the SCF coefficients of each of the derived atomic orbitals. The optimized exponents and SCF coefficients can then be used to derive suitable contraction exponents and contraction coefficients for a smaller basis set to be used in subsequent molecular calculations. 

Clinical use of lL-1, lL-6, and TNF- a, is limited by associated systemic toxicity. SCF seems better tolerated. It may be possible to develop derivatives of other cytokiaes that are less toxic. The synthetic nonapeptide VQGEESNDK (position 163—171 of human lL-1 P) (see Amino acids ... 

When deriving the Hartree-Fock equations it was only required that the variation of the energy with respect to an orbital variation should be zero. This is equivalent to the first derivatives of the energy with respect to the MO expansion coefficients being equal to zero. The Hartree-Fock equations can be solved by an iterative SCF method, and... 

Minimizing the total energy E with respect to the MO coefficients (see Refs. 2 and 3) leads to the matrix equation FC = SCE (where S is the overlap matrix). Solving this matrix is called the self-consistent field (SCF) treatment. This is considered here only on a very approximate level as a guide for qualitative treatments (leaving the more quantitative considerations to the VB method). The SCF-MO derivation in the zero-differential overlap approximations, where overlap between orbitals on different atoms is neglected, leads to the secular equation... 

Recently, a nonempirical rr-electron SCF approach was reported and applied to interpretations of spectra of various conjugated hydrocarbon radicals (147). The greatest attention, however, has been paid to radical ions derived from even alternant hydrocarbons (10, 58-60, 63, 125, 135, 148-153). Here, numerous experimental material suitable for systematic testing of the MO methods has been accumulated. In particular, the following sources of experimental data should be mentioned Hamill and collaborators (24) prepared... 

---