Variational methods for continuum states

Scattering by an N-electron atom or molecule with fixed nucleus is described by an (V +l)-electron Schrodinger wave function of the form 

The index 5 here denotes a particular degenerate solution, specified by boundary conditions, at given total energy E. The iV-electron function p is a target state with 

For an atom, or outside a sphere that completely encloses a target molecule, channel orbital functions are of the form 

Using the projection-operator formalism of Feshbach [ 115,116], an implicit variational solution for the coefficients cIJiS in can be incorporated into an equivalent partitioned equation for the channel orbital functions. This is a multichannel variant of the logic used to derive the correlation potential operator vc in orbital-functional theory. Define a projection operator Q such that 

Defining a Schrodinger functional S = (d H — the Euler-Lagrange equations for fixed energy E are 

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Variational methods for continuum states 8.4.1 Variational theory of the IZ-operator... 

Variational methods for continuum states Table 8.1. Partial wave phase shifts for He... 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

Recent work on using explicit waters in cluster-continuum oi implicit-explicit thermodynamic cycles shows much promise, as long as the standard state issues for water are consistent [37,41]. The key point is that water as a solvent, and water as a solute, and all species involved in the thermodynamic cycle, must be in a 1 M standard state. At this point it is not clear how many explicit waters should be used in a cycle [37], although use of the variational method to determine the number of waters to be used, and putting the waters together as clusters instead of monomers, appears to have much promise [41]. 

Previous studies [37, 38] of the doubly excited autoionizing states of H2 show that the lowest resonance is dominated by the s and d partial waves. Therefore, we use three Is-cSTO-fVGs and three 3d-cSTO-A/Gs. As discussed in Introduction, the selection of complex orbital exponents is not an easy task at all, and in this work, we attempt to propose a systematic way to select the orbital exponents for cSTOs. A hint for the selection can be obtained by relating the CBF method to the complex Kohn variation method [22, 39]. In the latter method, the outgoing continuum wave function is represented as a linear combination of basis functions and a few non functions satisfying the outgoing asymptotic behavior. In the CBF method, only functions are used thus, additional basis functions need to... 

Table 8.1 Some of the data used to construct Fig. 8.2. Variation of energy with C1-/C distance for the SN2 reaction CP + CH3C1 in water. Calculations by the author using B3LYP/6-31+G with the continuum solvent method SM8 [22] as implemented in Spartan [31]. The r of the x-axis in Fig. 8.2 is rc cl — r(transition state) = rc ci — 2.426. Flartrees were converted to kJ moP1 by multiplying by 2,626...

The SES, ESP, and NES methods are particularly well suited for use with continuum solvation models, but NES is not the only way to include nonequilibrium solvation. A method that has been very useful for enzyme kinetics with explicit solvent representations is ensemble-averaged variational transition state theory [26,27,87] (EA-VTST). In this method one divides the system into a primary subsystem and a secondary one. For an ensemble of configurations of the secondary subsystem, one calculates the MEP of the primary subsystem. Thus the reaction coordinate determined by the MEP depends on the coordinates of the secondary subsystem, and in this way the secondary subsystem participates in the reaction coordinate. 

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