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Frontmatter

Hypervirial Theorems. Development and Applications of the Hypervirial Methodology to Solve Quantum Chemistry Models

Hypervirial Theorems. Development and Applications of the Hypervirial Methodology to Solve Quantum Chemistry Models

Abstract
There are a very few quantum mechanical models that can be exactly solv ed. The most important and well-known ones are perhaps the hydrogen atom and the harmonic oscillator. The Schrödinger equation for almost all the problems the theoretical chemists and physicists have to deal with cannot be solved in a closed way. Due to this, several approximate methods are currently used to obtain eigenvalues, eigenfunetions and expectation values of physical observables. The most widespread procedures are probably the perturbation theory and the variation method which are describ ed in all textbooks on quantum chemistry and quantum mechanics. Since 1960 the hypervirial theorems have become a very useful tool in handling quantum chemical and quantum mechanical problems. They are simple mathematical relationships, which are called hypervirial relationships, that the trial wavefunction should obey if it is supposed to be an acceptable approximation to the actual wavefunction.
Francisco M. Fernández, Eduardo A. Castro

Part A

I. Hypervirial Theorems and Exact Solutions of the Schrödinger Equation

Abstract
According to the postulates of quantum mechanics [1–5] the state of the system ψ(0) at t = 0 is related to the state of the system ψ (t) at any other time t through:
$$\psi \left( t \right) = U\left( t \right)\psi \left( 0 \right)$$
(1)
where U(t) Is an evolution operator. The reader interested In a rigorous mathematical treatment of the evolution operators and their properties is referred to Refs. 4 and 5. A comprehensive summary is given in Apendix I. It immediately follows from the properties of U(t) that
$$\psi \left( 0 \right) = U^ + \left( t \right)\psi \left( t \right)$$
(2)
where U is the adjoint of U.
F. M. Fernández, E. A. Castro

II. Hypervirial Theorems and Perturbation Theory

Abstract
In Appendix IV we present perturbation theory (PT) in such a way that degenerate systems may be studied without further difficulties and we show how the equations corresponding to the Rayleigh-Schrödinger Perturbation Theory (RSPT) [1] can be derived. Since this last methodology is considered in this chapter, we deem convenient to deduce here the main equations in a sketchy way.
F. M. Fernández, E. A. Castro

III. Hypervirial Theorems and the Variational Theorem

Abstract
In Chapter II we have dealt with one of the two most important methods that allow one to get approximations for the solutions of the Schrödinger equation, i.e. PT. The other relevant method is the variational approximation, which will be discussed briefly in this section.
F. M. Fernández, E. A. Castro

IV. Non Diagonal Hypervirial Theorems and Approximate Functions

Abstract
In the previous chapter we have determined which are the conditions to be satisfied by an approximate function to obey some DHT, but up to now we have not set forth the indispensable requirements, if existing, for a trial function fulfills some NDHR.
F. M. Fernández, E. A. Castro

V. Hypervirial Functions and Self-Consistent Field Functions

Abstract
Among all the variational functions, the so-called Self-Consistent Field (SCF) ones are the most popular in Quantum Chemistry because they are especially important to study many-particle systems. Regarding Chemistry these functions offer a great help to study atomic and molecular systems [1, 2], in solid state theory, etc. Recently, their applications have been extended to the field of Molecular Vibrations [3–6].
F. M. Fernández, E. A. Castro

VI. Perturbation Theory without Wave Function

Abstract
We showed in section 9 how the RSPT allows one to obtain the energy and the wave function corrections via the resolution of some differential equations. Here we present a method that combines HR and PT and has proven to be extremely powerful when it is applied to simple models.
F. M. Fernández, E. A. Castro

Part B

VII. Importance of the Different Boundary Conditions

Abstract
The most usual quantum-mechanical models used in Theoretical Chemistry demand as boundary condition (BC) for bound states that the wave function and its first derivative must tend to infinite more quickly than any finite coordinate power. Notwithstanding, there are a large and important number of problems whose treatment requires finite BC, i.e. conditions imposed on the wave functions for finite coordinate values.
F. M. Fernández, E. A. Castro

VIII. Hypervirial Theorems for 1D Finite Systems. General Boundary Conditions

Abstract
The finite BC confront us with a problem no previously found in those cases studied in Part A. Let us suppose that ψi, ψj are two functions that obey the BC of the problem, so that they belong to DH. If ω is an arbitrary linear operator, then in general, ωψj. does not belong to DH.
F. M. Fernández, E. A. Castro

IX. Hypervirial Theorems for 1D Finite Systems. Dirichlet Boundary Conditions

Abstract
We have deduced the HT for some GBC. The Imposition of limiting conditions for A and B gives some particular BC, one of which will be discussed in this Chapter.
F. M. Fernández, E. A. Castro

X. Hypervirial Theorems for Finite 1D Systems. Von Neumann Boundary Conditions

Abstract
The treatment of the von Neumann Boundary Conditions (VNBC) will not be so detailed as that made for DBC, because simple examples to make a proper comparison there have not been reported in the current literature. However, some of the next theoretical results to be derived in what follows will be suggestible and interesting enough to deserve their examination.
F. M. Fernández, E. A. Castro

XI. Hypervirial Theorems for Finite Multidimensional Systems

Abstract
This chapter deals with the form that HT take as well as their practical utility for many-particle systems requiring different BC over arbitrary surfaces. Consequently, we will present here a generalization of previous results. Although the general aspects of the problem are the same, these models are more realistic and interesting than the 1D ones.
F. M. Fernández, E. A. Castro

Backmatter

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