Effective Schrodinger equation

The surrounding medium system (the m-system) can also formally be represented with an effective Schrodinger equation having the same form as Eq.(17). 

Projecting the Schrodinger equation onto the product space <8> Wp <8>M, we have an effective Schrodinger equation... 

The effective electronic Hamiltonian, /7eff, for the solute has already been introduced in the contribution by Tomasi. It describes the solute under the effect of the interactions with its environment and determines how these interactions affect the solute electronic wavefunction and properties. The corresponding effective Schrodinger equation reads... 

As said before, the nonlinear nature of the effective Hamiltonian implies that the Effective Schrodinger Equation (1.107) must be solved by an iterative process. The procedure, which represents the essence of any QM continuum solvation method, terminates when a convergence between the interaction reaction field of the solvent and the charge distribution of the solute is reached. 

Similar expressions and properties of the free energy functional (1.118) hold for all other levels of the QM molecular theory the factor is present in all cases of linear dielectric responses. More generally, the wavefunctions that make the free energy functional (1.117) stationary are also solutions of the effective Schrodinger Equation (1.107). 

The energy and wavefunction of the solvated solute molecule are obtained by solving the effective Schrodinger equation ... 

The electronic wavefunction of the solute, now in solution, can be obtained by solving the associated effective Schrodinger equation. 

Here, // is the in vacuo Hamiltonian of the structure X, and is the electron wavefunction of the structure X calculated in the presence of the perturbation due to the solvent. PX is obtained by solving the effective Schrodinger equation, Equation (4.141). Eb and Ea are calculated using the geometries optimized in solution. 

Reducing the degrees of freedom of the only nucleus is fruitful in the case of a heavy nucleus. In the positronium atom the nucleus has the same mass as the electron and it is useful to treat both particles symmetrically. It is well known that the a4m terms originate not only from relativistic effects, but also from annihilation contributions and the Fermi interaction. Due to that, the most useful approximation is a non-relativistic one and the final single-body equation is an effective Schrodinger equation with Coulomb interaction. This approach, based on an effective equation, was also developed for the few-body problem in nucleus physics. 

In both cases we can introduce a similar picture in terms of an effective Hamiltonian giving rise to an effective Schrodinger equation for the solvated solute. Introducing the standard Born-Oppenheimer approximation, the solute electronic wavefunction ) will satisfy the following equation ... 

For both methods, we describe the interactions between the quantum subsystem and the classical subsystem as interactions between charges and/or induced charges/dipoles and a van der Waals term [2-18]. The coupling between the quantum subsystem and the classical subsystem is introduced into the quantum mechanical Hamiltonian by finding effective interaction operators for the interactions between the two subsystems. This provides an effective Schrodinger equation for determining the MCSCF electronic wave function of the molecular system exposed to a classical environment, a structured environment, such as an aerosol particle. 

The effective Schrodinger equation of the pair in the sea is obtained by varying an independent slj with respect to u, but subject to >... 

Equations (152) and (153) in the previous section hold only for the pair functions which satisfy their respective effective Schrodinger equations (70) or (100a). To obtain the MO pairs Aij, one needed to minimize variational expressions like Eq. (86). In the same way, Eqs. (152b) and (153b) hold only for the optimum ju. To obtain these directly one must have a variational principle for each /ip,. 

The fact that we require the second solution of the effective Schrodinger equation is not due to the Schrodinger equation itself but due to the fact that we have implicitly made some rather gross model assumptions about the electronic structure of atoms in setting up this simple case. 

If we do this then both the form of and the local form of the pseudopotential are fixed since the latter contains y- In this case it is now worth looking briefly at the physical interpretation of the pseudopotential and the orbital x which are inextricably linked together in the local form. The orbital calculated by the solution of an effective Schrodinger equation is often called a pseudo-orbital to emphasise its origins and to remind us of its non-unique nature. 

We can now go back to the QM aspects of the PCM model by considering the methods for approximated solution of the effective non-linear Schrodinger equation for the solute. In principle, any variationedly approximated solution of the effective Schrodinger equation can be obtained by imposing that first-order variation of G with respect to an arbitrary vaxiar tion of the solute wavefunction is zero. This corresponds to a search of the minimum of the free energy functional within the domain of the variar tional functional space considered. In the case of the Hartree-Fock theory,... 

In the continuum approach to the surrounding medium one has by definition, m = 0. Medium effects are therefore presented by a reaction field term in Eq. (24). Three types of environment can be represented in this framework i) an anisotropic medium without spatial dispersion, where the permitivity tensor is defined with the ansatz c(r — r ) = c(r) (5(r — r ), that leads to a distance dependent dielectric system dependance ii) an isotropic medium which is characterized by s(r — r ) = e(r)1 (r — r ) hi) a homogeneous and isotropic medium, the permitivity tensor is the unit tensor multiplied by the static dielectric constant Thus the effective Schrodinger equation for each case is obtained from Eq. (24) after integration of the r -variable with the corresponding ansatz for the permitivity tensor. 

Minimization of Qtotlpu] leads to an effective Schrodinger equation where the Hamiltonian is ... 

Once an effective Hamiltonian for the system has been defined, the wave function T and energy can be evaluated by minimizing the Vsb in Eq. (8.6) with respect to the molecular orbital coefhcients of the ground state wave function using a self-consistent (SCF) procedure to solve the effective Schrodinger equation. 

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