Quantum chemistry variational principle

It has been said that mathematics is queen of the sciences. The variational branch of mathematics is essential both for understanding and predicting the huge body of observed data in physics and chemistry. Variational principles and methods lie in the bedrock of theory as explanation, and theory as a quantitative computational tool. Quite simply, this is the mathematical foundation of quantum theory, and quantum theory is the foundation of all practical and empirical physics and chemistry, short of a unified theory of gravitation. With this in mind, the present text is... 

The idea of coupling variational and perturbational methods is nowadays gaining wider and wider acceptance in the quantum chemistry community. The background philosophy is to realize the best blend of a well-defined theoretical plateau provided by the application of the variational principle coupled to the computational efficiency of the perturbation techniques. [29-34]. In that sense, the aim of these approaches is to improve a limited Configuration Interaction (Cl) wavefunction by a perturbation treatment. 

When the limiting conditions of the friction approximation are not valid, e.g., there is strong non-adiabatic coupling or rapid temporal variation of the coupling, there is at present no well-defined first principles method to calculate the breakdown in the BOA. The fundamental problem is that DFT cannot calculate excited states of adsorbates and quantum chemistry techniques, that can in principle calculate excited states, are not possible for extended systems. 

Before undertaking the major subject of variational principles in quantum mechanics, the present chapter is intended as a brief introduction to the extension of variational theory to linear dynamical systems and to classical optimization methods. References given above and in the Bibliography will be of interest to the reader who wishes to pursue this subject in fields outside the context of contemporary theoretical physics and chemistry. The specialized subject of optimization of molecular geometries in theoretical chemistry is treated here in some detail. 

In the one-dimensional search methods there are two principle variations some methods employ only first derivatives of the given function (the gradient methods), whereas others (Newton s method and its variants) require explicit knowledge of the second derivatives. The methods in this last category have so far found very limited use in quantum chemistry, so that we shall refer to them only briefly at the end of this section, and concentrate on the gradient methods. The oldest of these is the method of steepest descent. 

As we have noted, electronic stracture techniques attempt to solve the Schrodinger equation. The traditional approach in quantum chemistry has been to use the Hartree Fock (HF) approximation, in which a determinantal, antisymmetrized wave function is optimized in accordance with the variational principle. The wave function is normally written as an expansion of atomic orbitals (the LCAO approximation). A major weakness of the HF method is that in its single... 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

As is often the case in any new development in quantum chemistry, the fundamental formalism has been around for some time, in this case TDCPHF based on Frenkel s variational principle and density matrices [7]. However, the detailed theory will not be our immediate concern, but rather how it has been put to use. 

We have thus been able to construct a wave function that describes the qualitative behavior of the electronic stmcture for all internuclear distances. The price we have paid is to leave the single configurational description and construct the wave function as a linear combination of several configurations (determinants) with expansion coefficients to be determined by the variational principle together with the molecular orbital coefficients. This is the multiconfigurational approach in quantum chemistry. Before we end this section let us take a look at a more complex chemical bond, that in the Cr2 molecule. 

Since its eigenvalues correspond to the allowed energy states of a quantum-mechanical system, the time-independent Schrodinger equation plays an important role in the theoretical foundation of atomic and molecular spectroscopy. For cases of chemical interest, the equation is always easy to write down but impossible to solve exactly. Approximation techniques are needed for the application of quantum mechanics to atoms and molecules. The purpose of this subsection is to outline two distinct procedures—the variational principle and perturbation theory— that form the theoretical basis for most methods used to approximate solutions to the Schrodinger equation. Although some tangible connections are made with ideas of quantum chemistry and the independent-particle approximation, the presentation in the next two sections (and example problem) is intended to be entirely general so that the scope of applicability of these approaches is not underestimated by the reader. 

Eqs. (l)-(3), (13), and (19) define the spin-free CGWB-AIMP relativistic Hamiltonian of a molecule. It can be utilised in any standard wavefunction based or Density Functional Theory based method of nonrelativistic Quantum Chemistry. It would work with all-electron basis sets, but it is expected to be used with valence-only basis sets, which are the last ingredient of practical CGWB-AIMP calculations. The valence basis sets are obtained in atomic CGWB-AIMP calculations, via variational principle, by minimisation of the total valence energy, usually in open-shell restricted Hartree-Fock calculations. In this way, optimisation of valence basis sets is the same problem as optimisation of all-electron basis sets, it faces the same difficulties and all the experience already gathered in the latter is applicable to the former. 

Explicitly correlated wave function fheory [14] is anofher imporfanf approach in quantum chemistry. One introduces inter-electron distances together with the nuclear-electron distances and set up some presumably accurate wave function and applies the variation principle. The Hylleraas wave function reported in 1929 [15] was the first of this theory and gave accurate results for the helium atom. Many important studies have been published since then even when we limit ourselves to the helium atom [16-28]. They clarified the natures and important aspects of very accurate wave functions. However, the explicitly correlated wave function theory has not been very popularly used in the studies of chemical problems in comparison with the Hartree-Fock and electron correlation approach. One reason was that it was generally difficult to formulate very accurate wave functions of general molecules with intuitions alone and another reason was that this approach was rather computationally demanding. 

There are lots of exehange-eorrelation potentials in the literature. There is an impression that their authors worried most about theory/experiment agreement. We ean hardly admire this kind of seienee, but the alternative (i.e., the practiee of ab initio methods with the intact and holy Hamiltonian operator) has its own disadvantages. This is because finally we have to choose a given atomic basis set, and this influences the results. It is true that we have the variational principle at our disposal, and it is possible to tell which result is more accurate. But more and more often in quantum chemistry, we use some non-variational methods (cf. Ch ter 10). Besides, the Hamiltonian holiness disappears when the theory becomes relativistic (cf. Qiapter 3). 

For molecules, Hartree-Fock approximation is the central starting point for most ab initio quantum chemistry methods. It was then shown by Fock that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, has the same antisymmetric property as the exact solution and hence is a suitable ansatz for applying the variational principle. 

In the limit of a complete basis set, this equation becomes equivalent to the Schrodinger equation. For a finite basis set, Equation (2) represents the best wave function (in the sense of the variation principle) that can be obtained. It is called the Full Cl (FCI) wave function. It serves as a calibration point for all approximate wave-function methods. It is obvious that many of the coefficients in Equation (3) are very small. We can consider most approximate MO models in quantum chemistry as approximations in one way or the other, where one attempts to include the most important of the configurations in Equation (2). We notice that the FCI wave function and... 

How severe the problem might be has been shown by M. Stanke, J. Karwowski, Variational Principle in the Dirac Theory Spurious Solutions, Unexpected Extrema and Other Thaps in New Trends in Quantum Systems in Chemistry and Physics , vol. I, p. 175-190, eds. J. Maruani et at, Kluwer Academic Publishers. Sometimes an eigenfunction corresponds to a quite different eigenvalue. Nothing of that sort appears in non-relativistic calculations. 

Komi, D. 1., T. Markovich, N. Maxwell, and E. R. Bittner. 2009. Supersymmetric quantum mechanics, excited state energies and wave functions, and the Rayleigh-Ritz variational principle a proof of principle study. Journal of Physical Chemistry A 113 (52) 15257. 

The variation principle is used to get the best possible approximation of a function. In almost all applications of quantum chemistry, the coefficients are optimized in a linear expansion. In Chapter 2, we will use the variation principle to derive the Hartree-Fock equation. 

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