Self-consistent field method separability

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

In order to find a good approximate wave function, one uses the Hartree-Fock procedure. Indeed, the main reason the Schrodinger equation is not solvable analytically is the presence of interelectronic repulsion of the form e2/r. — r.. In the absence of this term, the equation for an atom with n electrons could be separated into n hydrogen-like equations. The Hartree-Fock method, also called the Self-Consistent-Field method, regards all electrons except one (called, for instance, electron 1), as forming a cloud of electric charge... 

R.B. Gerber and M.A. Ratner, Self-consistent field methods for vibrational excitations in polyatomic molecules, Adv. Chem. Phys., 70 (1988) 97. P. Jungwirth and R.B. Gerber, Quantum dynamics of large polyatomic systems using a classically based separable potential method, J. Chem. Phys., 102 (1995) 6046 Quantum dynamics of many atom systems by classically based separable potential (CSP) method Calculations for T (Ar),2 in full dimensionality, J. Chem. Phys., 102 (1995) 8855. 

Consider, as an example, polyethylene if the chains are packed as in the crystal (Fig. 1), there are 5.5 x 10 per m in the plane perpendicular to their length. Hence the tensile fracture stress would be 33 GPa. Variants of this approach, using simple Morse potentials, provide values ranging from about 19 to 36 GPa. Other calculations using Hartree-Fock self-consistent field methods yield values of 66 GPa at 0 K (Crist et al., 1979). This last value seems a bit high and may be due to inaccuracies of the Hartree-Fock approximation at large atomic separations. 

The standard way of treating conjugated systems using the o-n separation approximation in the 1930s and into the 1960s was the so-called Hiickel methody In this method, the electron-electron interactions are not explicitly considered. Rather, the positions of the nuclei are fixed, and the electrons move in the field of the nuclei. Much of the error that results from neglecting the interactions of the electrons with one another can be circumvented with proper adjustment of empirical parameters. These parameters are also adjusted to allow (approximately) for the interaction from the o system, which is thus taken into account without specific calculations. This method is crude but does often give qualitative results that enable rationalization of many chemical phenomena of interest. It was a powerful and useful tool in its time. A better approximation is the Hartree-Fock or self-consistent field (SCF) method in which the electron-electron interactions are explicitly considered (Self-Consistent Field Method in Chapter 3). The quantum mechanical calculations on the n system in this case are carried out in a... 

For planar unsaturated and aromatic molecules, many MO calculations have been made by treating the a and n electrons separately. It is assumed that the o orbitals can be treated as localized bonds and the calculations involve only the tt electrons. The first such calculations were made by Hiickel such calculations are often called Hiickel molecular orbital (HMO) calculations Because electron-electron repulsions are either neglected or averaged out in the HMO method, another approach, the self-consistent field (SCF), or Hartree-Fock (HF), method, was devised. Although these methods give many useful results for planar unsaturated and aromatic molecules, they are often unsuccessful for other molecules it would obviously be better if all electrons, both a and it, could be included in the calculations. The development of modem computers has now made this possible. Many such calculations have been made" using a number of methods, among them an extension of the Hiickel method (EHMO) and the application of the SCF method to all valence electrons. ... 

In the MQC mean-field trajectory scheme introduced above, all nuclear DoF are treated classically while a quantum mechanical description is retained only for the electronic DoF. This separation is used in most implementations of the mean-field trajectory method for electronically nonadiabatic dynamics. Another possibility to separate classical and quantum DoF is to include (in addition to the electronic DoF) some of the nuclear degrees of freedom (e.g., high frequency modes) into the quantum part of the calculation. This way, typically, an improved approximation of the overall dynamics can be obtained—albeit at a higher numerical cost. This idea is the basis of the recently proposed self-consistent hybrid method [201, 202], where the separation between classical and quantum DoF is systematically varied to improve the result for the overall quantum dynamics. For systems in the condensed phase with many nuclear DoF and a relatively smooth distribution of the electronic-vibrational coupling strength (e.g.. Model V), the separation between classical and quanmm can, in fact, be optimized to obtain numerically converged results for the overall quantum dynamics [202, 203]. 

Continued Quantum and Molecular Mechanical Simulations, In this technique, a molecular dynamics simulation includes the treatment of some part of the system wilh a quantum mechanical technique. This approach. yMf.MM. is similar to programs that Use quantum mechanical methods to treat the n-systems of the structures in question separately from the sigma framework. The results are combined ai ihe end to render a slructure which is optimized and energy-refined in satisfy both self-consistent field (SCF) and force field energy convergence. 

The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. 

Another, purely quantum mechanical approximation is the so-called time-dependent self-consistent field (TDSCF) method. For general reviews see Kerman and Koonin (1976), Goeke and Reinhard (1982), and Negele (1982). For applications to molecular systems see, for example, Gerber and Ratner (1988a,b). In the TDSCF method the wavepacket is separated according to... 

For the majority of enzyme-catalysed reactions, covalently bonded parts of the system must be separated into QM and MM regions. There has been considerable research into methods for QM/MM partitioning of covalently bonded systems. Important methods include the local self-consistent field (LSCF) method,114115 and the generalized hybrid orbital (GHO) technique.116 Alternatively a QM atom (or QM pseudo-atom) can be added to allow a bond at the QM/MM frontier for example, the link atom method or the connection atom method. 

Another strategy to separate the QM part from the MM one is to freeze the pair of electrons in the broken bond (assumed to be a single bond). This has been suggested first by Warshel and Levitt [22] and the method has been developed recently at the semiempirical [23,241 and ab initio levels [26-281 as the local self-consistent field (LSCF) method. 

A different approach to treat correlation effects which are not well described within the LSDA consists in incorporating self-interaction corrections (SIC) [111-114] in electron structure methods for solids, Svane et al. [115-120]. In the Hartree-Fock (HF) theory the electron-electron interactions are usually divided into two contributions, the Coulomb term and the exchange term although they both are Coulomb interactions. The separation though, is convenient because simplifications of self-consistent-field calculations can be obtained by including in both terms the interaction of the electron itself. In the HF theory this has no influence on the solutions because these selfinteractions in the Coulomb and exchange terms exactly cancel each other. However, when the exchange term is treated... 

---