Hamiltonian effective hybrid

For the simplest case of a system partitioned into three regions, the effective hybrid Hamiltonian is composed of four terms ... 

The inter/intramolecular potentials that have been described may be viewed as classical in nature. An alternative is a hybrid quantum-mechanical/classical approach, in which the solute molecule is treated quantum-mechanically, but interactions involving the solvent are handled classically. Such methods are often labeled QM/MM, the MM reflecting the fact that classical force fields are utilized in molecular mechanics. An effective Hamiltonian Hefl is written for the entire solute/solvent system ... 

In the frame of the target hybrid QM/MM procedure, only the electronic structure of the R-system is calculated explicitly. For this reason, we consider its effective Hamiltonian eq. (1.235) in more detail. It contains the operator terms coming from (1) the Coulomb interaction of the effective charges in the M-system with electrons in the R-system 5VM and (2) from the resonance interaction of the R- and M-systems. 

In the frame of the hybrid methods it must be computed by a QM method. The Schrodinger equation with the effective Hamiltonian Hjf has multiple solutions, which describe excited states of the R-system provided the M-system is frozen in its ground state. Electronic energy of the system in the state expressed by the wave function eq. (1.231), has the form [29,30] ... 

Further elaboration of the hybrid models stipulated by the necessity to model chemical processes in polar solvents or in the protein environment of enzymes, or in oxide-based matrices of zeolites, requires the polarization of the QM subsystem by the charges residing on the MM atoms of the classically treated solvent, or protein, or oxide matrix. This polarization is described by renormalizing the one-electron part of the effective Hamiltonian for the QM subsystem ... 

If these variations are taken into account in the calculations on the QM part of the complex system, the effect of the MM system on the parameters of the effective Hamiltonian for the QM part turns out to be taken into account in the first order. It should be stressed that changes in the hybridization of the frontier atom due to participation of one orbital in the QM subsystem are not taken into account in any of the existing QM/MM schemes. This effect is not very large, so the first-order correction for taking it into account seems to be adequate. 

The plan of the chapter is as follows. Section 2 outlines why the use of hybrid potentials is important if large systems are to be studied and section 3 desribes the basic form of the effective Hamiltonian for a hybrid potential. The next four sections, sections 4 to 7, each cover a separate term in the hybrid Hamiltonian and section 8 describes the implementation of the hybrid potentials for use in actual calculations. Section 9 concludes. 

A rigorous derivation of the form of the hybrid potentials treated in this chapter requires the construction an effective Hamiltonian, Her, for the system. This Hamiltonian can then be used as the Hamiltonian for the solution of the time-independent Schrodinger equation for the wavefunc-tion of the electrons on the QM atoms, P, and for the potential energy of the system, qm/mm- If R are the coordinates of the MM atoms, the Schrodinger equation (equation 1) becomes ... 

The interaction Hamiltonian for many hybrid potentials contains a term which accounts for the electrostatic interaction between the MM atoms and the QM electrons and nuclei and a Lennard-Jones term that mimics the effects of the exchange repulsion and dispersion interactions between the MM and QM atoms. For an ab initio QM method, it takes the form ... 

In the case of hydrocarbons, the calculation still comprises five parameters, namely two Coulomb integrals ac and h, two bonds, resonance integrals /Sec and /Sqh and one resonance integral for two orbitals centered on the same carbon Ice- As usual, the interaction terms between non-bonded atoms are neglected. The parameters a and /3 are the matrix elements of a non-specified effective Hamiltonian with respect to the sp3 or sp2 hybrid orbitals of carbon and the Is orbitals of hydrogens. For the a bonds of conjugated hydrocarbons 4S>, the following set of values has been used... 

Tokmachev AM, Tchougreeff AL, Misurkin IA. Effective electronic Hamiltonian for quantum subsystem in hybrid QM/MM method as derived from APSLG description of electronic structure of classical part of molecular system. J Mol Struct Theochem 2000 506 17-34. 

As molecular applications of the extended DK approach, we have calculated the spectroscopic constants for At2 equilibrium bond lengths (RJ, harmonic frequencies (rotational constants (B ), and dissociation energies (Dg). A strong spin-orbit effect is expected for these properties because the outer p orbital participates in their molecular bonds. Electron correlation effects were treated by the hybrid DFT approach with the B3LYP functional. Since several approximations to both the one-electron and two-electron parts of the DK Hamiltonian are available, we dehne that the DKnl -f DKn2 Hamiltonian ( 1, 2= 1-3) denotes the DK Hamiltonian with DKnl and DKn2 transformations for the one-electron and two-electron parts, respectively. The DKwl -I- DKl Hamiltonian is equivalent to the no-pair DKwl Hamiltonian. For the two-electron part the electron-electron Coulomb operator in the non-relativistic form can also be adopted. The DKwl Hamiltonian with the non-relativistic Coulomb operator is denoted by the DKwl - - NR Hamiltonian. 

The steepest descent method is very effective far from the minimum of , but is always much less efficient than the Gauss-Newton method near the minimum of . Marquardt (1963) has proposed a hybrid method that combines the advantages of both Gauss-Newton and steepest descent methods. Mar-quardt s method, combined with the Hellmann-Feynman pseudolinearization of the Hamiltonian energy level model, is the method of choice for most nonlinear molecular spectroscopic problems. 

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