Energy linear variation theory

There turns out to be a way to determine not only the energy but also the coefficients. This powerful use of variation theory is called linear variation theory. 

The above determinant is called a secular determinant. Linear variation theory rests on equation 12.31 if the secular determinant formed from the energy and overlap integrals and the energy eigenvalues (which are the unknowns ) is equal to zero, then the equations 12.30 will be satisfied and the energy will be minimized. 

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. 

The next set of open-shell cluster expansion theories to appear on the scene emphasized the size-extensivity feature (al), and all of them were designed to compute energy differences with a fixed number of valence electrons. Several related theories may be described here - (i) the level-shift function approach in a time-dependent CC framework by Monkhorst/56/ and later generalizations by Dalgaard and Monkhorst/57/, also by Takahasi and Paldus/105/, (ii) the CC-based linear response theory by Mukherjee and Mukherjee/58/, and generalized later by Ghosh et a 1/59.60.107/,(iii)the closely related formulations by Nakatsuji/50,52/ and Emrich/62/ and (iv) variational theories by Paldus e t a I / 54/ and Saute et. al /55/ and by Nakatsuji/50/. 

Figure 47 shows the qualitative behavior of this free energy density. A crucial feature is that the renormalized distance xR corresponds still to the inverse scattering intensity S-l(q) at q = q. Since xocxocl/T in simple polymers, the nonlinear relation between x and xR then implies a nonlinear relation between xR and 1/T. Thus while Leibler s theory [43] predicts a linear variation of S" (q ) with 1/T (near the temperature where S-1(q ) should vanish for f = 1/2), the fluctuation effects of Helfand and Fredrickson [58] imply a curved variation of S l(q ) with 1/T. Such a curved variation indeed is found both in experimental data [317-323] and simulations [325, 328], see Figs. 43b, 48. Of course, due to finite size problems in the simulation one cannot as yet detect the small jump singularity that signals the mesophase separation transition in the experiment (Fig. 48). 

For the van der Waals component no such analytical theory exists. Aqvist and co-workers assumed that a similar linear treatment would work for these interactions but with a different empirical factor, to be determined from calibration experiments. There was some indirect evidence that this approach would be reasonable. For example, the experimental free energies of solvation for various hydrocarbons (e.g. n-alkanes) depend in an approximately linear fashion on the length of the carbon chain. In addition, the mean van der Waals solute-solvent energies from molecular dynamics simulations did show a linear variation with chain length (the slope of the line varying according to the solvent). 

Having now discussed how one can go about optimizing the electronic energy of an MCSCF wavefunction, we turn our attention to two special subclasses of this procedure the single-configuration SCF problem and the frozen-orbital Cl problem. Because we choose to view these situations as special cases of the above MCSCF problem, we obtain a specialized view of SCF and Cl theory. There already exist in the literature extensive and clear treatments of SCF and Cl as they are more commonly treated within the linear variational framework. Hence we have not attempted to cover the more conventional aspects of these topics here. 

Linear response theory expression Alternatively, the spin-spin coupling constant can be expressed using the linear response theory formalism. Let us write the electronic energy of the system perturbed by the nuclear magnetic dipole moments M/f in the form E = E(Mjf, A), where A are the variational parameters of the wave function. A may represent orbital rotation parameters for the SCF wave function, or orbital rotation parameters and coefficients of the configuration interaction expansion for the MCSCF... 

Since in MCRPA the reference wavefunction is a variational MCSCF wave-function, one can derive the MCRPA also by application of linear response theory, Section 11.2, or of the quasi-energy derivative method, Section 12.3, to this MCSCF state. 

To obtain an optimum it may be desirable to employ the variational theory for optimizing the total energy of the system with respect to any deformation under certain constrained conditions. Since highly crosslinked network systems are fairly "rigid," without losing the generality, they may be considered to deform linearly elastic. This is particularly true if deformation is small. 

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