Exact and Approximate Wave Functions

We have now developed the basic tools needed for the construction of approximate electronic wave functions and for analysing the different approaches to molecular electronic-structure calculations. However, before we embark on this project, it is useful to consider in general terms some of the requirements we would like to place on any approximate wave function. We therefore discuss in this chapter the relationship between exact wave functions and approximate wave functions, with emphasis on topics such as size-extensivity, the variation principle and symmetry restrictions. 

We first survey, in Section 4.1, the more important characteristic properties of the exact solution to the time-independent Schrodinger equation for a molecular electronic system and relate these characteristics to those of approximate wave functions. More detailed treatments of some of these topics follow in the subsequent sections the variation principle in Section 4.2, size-extensivity in Section 4.3 and symmetry constraints in Section 4.4. 

---

We have to stress that these eonclusions will be valid independent of the approximations used to eompute the molecular wave functions. The reason is that they follow from the symmetry, which is identical for the exact and approximate wave functions. 

We consider in this section the variation principle in molecular electronic-structure theory. Having established the particular relationship between the Schrddinger equation and the variational condition that constitutes the variation principle, we proceed to examine the variation method as a computational tool in quantum chemistry, paying special attention to the application of the variation method to linearly expanded wave functions. Next, we examine two important theorems of quantum chemistry - the Hellmann-Feynman theorem and the molecular virial theorem - both of which are closely associated with the variational condition for exact and approximate wave functions. We conclude this section by presenting a mathematical device for recasting any electronic energy function in a variational form so as to benefit to the greatest extent possible from the simplifications associated with the fulfilment of the variational condition. 

The first term is referred to as the diamagnetic contribution, while the latter is the paramagnetic part of the magnetizability. Each of the two components depend on the selected gauge origin however, for exact wave functions these cancel exactly. For approximate wave functions this is not guaranteed, and as a result the total property may depend on where the origin for the vector potential (eq. (10.61)) has been chosen. 

Two molecules, when isolated (say at infinite distance), are independent and the wave function of the total system might be taken as a product of the wave functions for the individual molecules. When the same two molecules are at a finite distance, then any product-like function represents only an approximation (sometimes a very poor one ), because according to a postulate of quantum mechanics, the wave function has to be antisymmetric with respect to the exchange of electronic labels, while the product does not fulfill that. More exactly, the approximate wave function has to belong to the irreducible representation of the symmetry group of the Hamiltonian (see Appendix C available at booksite.elsevier.com/978-0-444-59436-5, p. el7), to which the ground-state wave function belongs. This means, first of all, that the Pauli exclusion principle is to be satisfied. 

Ichikawa K, Wagatsuma A, Kusumoto M, Tachibana A (2010) Electronic stress tensor of the hydrogen molecular ion comparison between the exact wave fimctimi and approximate wave functions using Gaussian basis sets. J Mol Struct Theochem 951 49-59 Tachibana A (2010) Energy density concept a stress tensor approach. J Mol Struct Theochem 943 138-151... 

We cannot solve the Schroedinger equation in closed fomi for most systems. We have exact solutions for the energy E and the wave function (1/ for only a few of the simplest systems. In the general case, we must accept approximate solutions. The picture is not bleak, however, because approximate solutions are getting systematically better under the impact of contemporary advances in computer hardware and software. We may anticipate an exciting future in this fast-paced field. 

In such cases the expression from fii st-order perturbation theory (10.18) yields a result identical to the first derivative of the energy with respect to A. For wave functions which are not completely optimized with respect to all parameters (Cl, MP or CC), the Hellmann-Feynman theorem does not hold, and a first-order property calculated as an expectation value will not be identical to that obtained as an energy derivative. Since the Hellmann-Feynman theorem holds for an exact wave function, the difference between the two values becomes smaller as the quality of an approximate wave function increases however, for practical applications the difference is not negligible. It has been argued that the derivative technique resembles the physical experiment more, and consequently formula (10.21) should be preferred over (10.18). 

This proof shows that any approximate wave function will have an energy above or equal to the exact ground-state energy. There is a related theorem, known as MacDonald s Theorem, which states that the nth root of a set of secular equations (e.g. a Cl matrix) is an upper limit to the n — l)th excited exact state, within the given symmetry subclass. In other words, the lowest root obtained by diagonalizing a Cl matrix is an upper limit to the lowest exact wave functions, the 2nd root is an upper limit to the exact energy of the first excited state, the 3rd root is an upper limit to the exact second excited state and so on. 

The fundamental object in the quantum theory of matter is the wave function, which is the most compact way to represent all the information contained in a system. Exact wave functions are usually not available, so if we want to know certain properties of the system the procedure is to set up some model Hamiltonian and get an approximate wave function, from which the desired properties can be extracted. This program can be represented by... 

We have just explained that the wave equation for the helium atom cannot be solved exacdy because of the term involving l/r12. If the repulsion between two electrons prevents a wave equation from being solved, it should be clear that when there are more than two electrons the situation is worse. If there are three electrons present (as in the lithium atom) there will be repulsion terms involving l/r12, l/r13, and l/r23. Although there are a number of types of calculations that can be performed (particularly the self-consistent field calculations), they will not be described here. Fortunately, for some situations, it is not necessary to have an exact wave function that is obtained from the exact solution of a wave equation. In many cases, an approximate wave function is sufficient. The most commonly used approximate wave functions for one electron are those given by J. C. Slater, and they are known as Slater wave functions or Slater-type orbitals (usually referred to as STO orbitals). 

This method is usually thought as an approach allowing one to find the exact exchange potential. It may be considered [17] as an approximation to the exact GS problem, similar to the HF approximation namely, the solution of the optimized potential (OP) approximation - the energy Egg and the wave function Gs - stems from the following minimization problem... 

Establishing a hierarchy of rapidly converging, generally applicable, systematic approximations of exact electronic wave functions is the holy grail of electronic structure theory [1]. The basis of these approximations is the Hartree-Fock (HF) method, which defines a simple noncorrelated reference wave function consisting of a single Slater determinant (an antisymmetrized product of orbitals). To introduce electron correlation into the description, the wave function is expanded as a combination of the reference and excited Slater determinants obtained by promotion of one, two, or more electrons into vacant virtual orbitals. The approximate wave functions thus defined are characterized by the manner of the expansion (linear, nonlinear), the maximum excitation rank, and by the size of one-electron basis used to represent the orbitals. 

---