Electron Correlation in Molecules

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1

MOLECULAR ELECTRONIC STRUCTURE

1.1 Introduction

The calculation of properties of atoms and molecules from first principles has long been recognized to be of central importance in chemistry and in atomic and molecular physics. Theoretical studies enable the properties of species, which are difficult or indeed impossible to examine experimentally, to be determined by approximate solution of the appropriate quantum mechanical equations. Many compounds are too reactive to be isolated and cannot be studied by means of standard laboratory techniques, such as X-ray crystallography or infra-red spectroscopy. The dipole moment of a charged species cannot be measured experimentally and that of a free radical only with difficulty. However, such molecular properties can often be calculated quite accurately from first principles. (Calculated dipole moments do depend, of course, on the choice of origin.) Radicals and ions have been identified in the interstellar medium on the basis of their calculated properties (Wilson 1980d). These species have a very short lifetime in the terrestrial laboratory whereas in space, since collisions are infrequent, their lifetimes are much longer. Some chemical reactions take place in times of the order of a picosecond and experimentally the transition states can only be inferred indirectly by, for example, the effects of substituents on the reaction rate. Theoretical studies can provide valuable information about transition states.

In addition to providing a route to data for a particular molecule or reaction which are not available from experimental studies, theoretical chemistry can also lead to simple and useful chemical concepts which can then be used to rationalize vast quantities of data. For example, orbital theories have led to important models in chemical spectroscopy such as Koopmans’ theorem (Koopmans 1933) and in structural chemistry such as lone pairs and bonding pairs (Gillespie and Nyholm 1957). The importance of this interpretive aspect of theoretical chemistry was emphasized by C. A. Coulson (1973) when he remarked upon the futility of obtaining accurate numbers, whether by computation or by experiment, unless these numbers can provide us with simple and useful chemical concepts; otherwise one might as well be interested in a telephone directory. Theories of electron correlation in molecules must not only afford high accuracy but must also be open to simple qualitative interpretation.

The use of a theoretical rather than an experimental approach to the determination of the properties of molecules can increasingly be justified on economic grounds. Present-day computer programs are usually very flexible and the treatment of a different state or system or different properties of a molecule merely requires a change of the input data. For example, in the field of quantum pharmacology (Richards 1983) properties of molecules which are believed to have potential applications as drugs are calculated, thereby possibly avoiding the painstaking preparation and testing of hundreds of similar molecules before the required pharmacological activity can be found.

The first step in most approaches to the electronic structure of molecules is to decompose the N-electron problem into N one-electron problems. This leads to what are known as independent electron models, or, more simply, orbital models. Orbital models, such as the Hartree–Fock approximation, account for the majority, typically 99.5%, of the total energy of an atom or a molecule. It is unfortunate that the error in such orbital theories is of the same order of magnitude as most energies of chemical interest: binding energies, ionization energies, activation barriers, and the like. This error arises from the correlation of the motions of the individual electrons and this most crucial area is the topic to which this book is addressed.

Of course, the nature of electron correlation effects will depend to a large extent on the particular orbital model with respect to which they are computed. Electron correlation effects are not directly observable; it is only by considering both the results obtained within an orbital model and its correlation corrections that quantities can be obtained which can be compared with data derived from experiment. For this reason Chapter 2 is devoted to independent electron models. In Chapter 3, qualitative and some quantitative aspects of the electron correlation problem in molecules are discussed. Fundamental to modem treatments of electron correlation is the linked diagram theorem and this is addressed in Chapter 4. Powerful methods for the analysis and simplification of molecular correlation calculations can be obtained by employing group theoretical techniques. Group theoretical aspects of the correlation problem are discussed in Chapter 5. In general, molecular calculations can only be rendered tractable by employing the algebraic approximation; that is the orbitals are parameterized by expansion in a finite basis set. The algebraic approximation is discussed in detail in Chapter 6. In Chapter 7, many of the techniques currently being employed in the treatment of electron correlation effects are discussed. These techniques will be analysed in terms of the linked diagram theorem of Chapter 4. and the use of the group theoretical methods described in Chapter 5 will be indicated.

In this first chapter we aim to provide a brief outline of basic molecular physics and quantum chemistry. More detailed discussions can be found elsewhere (e.g. Messiah 1967; Ziman 1969; McWeeny and Sutcliffe 1976; Atkins 1983; Coulson 1961; Dirac 1958; McWeeny 1982; Landau and Lifshitz 1958). In Section 1.2, the Born–Oppenheimer approximation is discussed. This approximation is made in most applications of quantum mechanical methods to molecules. The electronic Schrödinger equation is introduced in Section 1.3. The electronic energy is a function of the nuclear coordinates and gives rise to potential energy curves and surfaces which are the subject of Section 1.4. In Section 1.5, the relationship between quantities which can be calculated and experimental observables is described briefly. The density matrix can provide a useful interpretation of molecular electronic structure and, therefore, this is discussed in Section 1.6. The majority of contemporary quantum mechanical calculations on molecules are performed within the framework of non-relativistic theory. Exact solution of the non-relativistic quantum mechanical equations for molecules leads to the so-called non-relativistic limit and this is the subject of Section 1.7. Finally, relativistic effects are very briefly discussed in Section 1.8.

1.2 The Born–Oppenheimer approximation

The separation of the nuclear and the electronic motion is almost invariably the first step in any application of quantum mechanics to molecules.

For a conservative system, our problem is the solution of the eigenvalue problem

, which completely defines the dynamical state of the system, and Ei is an allowed energy value. Within the framework of non-relativistic quantum mechanics, the total molecular hamiltonian for a system of N electrons and M nuclei takes the form

with nuclear kinetic energy

where MA is the mass of nucleus A, electronic kinetic energy

a nucleus–nucleus repulsion term

FIG. 1.1. Coordinate system. A, B,… denote the nuclei and i, j,… denote the electrons.

where ZA is the charge of nucleus A, the electron–electron repulsion term

and, finally, an electron–nucleus attraction term

Atomic units, which are defined in the Note (p. xi), and the coordinate system defined in Fig. 1.1, have been used throughout.

Equation (1.2.1) is a second-order differential equation in 3(N+M) variables. The complexity of the problem can be reduced considerably by invoking the Born–Oppenheimer approximation (Born and Oppenheimer 1927). Physically, it is expected that the disparity of the mass of the electrons and the mass of the nuclei in a molecule will allow the electronic motion to accommodate almost instantaneously any change in the positions of the nuclei. Consequently, the electronic and nuclear motion can be treated separately to a very good approximation. The electrons experience the nuclei as fixed force centres and adiabatically follow any change in the nuclear positions. Conversely, the nuclei experience an average interaction with the electrons.

The nuclear kinetic energy, TN, is very much smaller than the other terms in eqn (1.2.2). It will be assumed in the present discussion that spin-related effects are small compared with the nuclear kinetic energy. (The reader is referred to the monograph by Kronig (1930) for a discussion of the situation when spin effects are not small.) If, in eqn (1.2.2) MA , then

where the electronic hamiltonian operator is given by

Since

the eigenvalues of the electronic hamiltonian may be determined for a particular value of the nuclear position vectors, Rare implicit functions of the nuclear geometry; that is

where R has been used to denote the nuclear coordinates. The term arising from repulsions between the nuclei may be treated as a simple additive constant. Equation (1.2.11) is the electronic Schrödinger equation; the problem of describing the electronic motion accurately is the topic of this monograph.

, form a complete orthonormal set and can, therefore, be used to expand the total molecular wave function, for both the electrons and the nuclei, as follows

where σi denotes the spin function for the ith electron in the system. The Schrödinger equation for the complete system, eqn (1.2.1), may then be written as

which may be rewritten as

Multiplying eqn and integrating over the coordinates of all the electrons leads to

It can be readily demonstrated that

and hence

Note that the second term on the right-hand side of this equation is zero when m = n. We have

where

and

Substituting eqn (1.2.18), together with eqns (1.2.19–21), into eqn (1.2.15), we obtain

where

In the adiabatic approximation (see, for example, Kolos 1970), it is assumed that the Cs can be neglected giving

or

which is the nuclear eigenvalue equation in which the ENn is an effective potential for the nuclei. In the Born–Oppenheimer approximation (Born and Oppenheimer 1927; Born and Huang 1954) in addition to taking the Cs to be zero it is assumed that the effective potential for the nuclei is equal to the electronic energy function, that is ENn(R) = En(R).

It should be noted that the separation of the nuclear and electronic motion described above is only valid as long as there is no significant coupling between different electronic states induced by the nuclear motion. The Born–Oppenheimer approximation is valid only if the energy levels are not degenerate and there are no nearby levels of the same symmetry.

The Born–Oppenheimer approximation is found to be excellent in practice. Only in the most accurate quantum mechanical treatments of molecules containing light atoms are errors arising from this approximation significant (see, for example, the calculations on the hydrogen molecule by Kolos et al. 1960, 1963, 1964).

1.3 Molecular electronic structure

The electronic structure of a molecule is defined, within the Born–Oppenheimer approximation, as the solution of the Schrödinger equation for the motion of the electrons in the field of fixed nuclei

Although the Born–Oppenheimer approximation simplifies the quantum mechanics of molecules considerably, further approximations are necessary in order to develop theoretical machinery which leads to computationally tractable algorithms. This monograph is concerned with development of methods for performing accurate calculations of the electronic structure of molecules, however, it is important to remember that any theory, no matter how sophisticated, will contain errors, although for useful models these errors will be small.

Molecular calculations are usually performed for one of two reasons:

(i) Testing a new theory or algorithm;

(ii) Complementing, verifying, or interpreting information derived from experiment about a particular molecular system.

Most of the molecular calculations which will be considered in this monograph are concerned with the former aspect; the demonstration of various properties of different approaches to the correlation problem in molecules. However, in this section, we wish to emphasize that some care must be exercised in employing any theory, but particularly a recently developed theory, in studies of molecules performed for the second of the two reasons given above.

In practical applications of the methods of theoretical chemistry there is much advantage in studying a wide range of problems at a uniform level of approximation (see, for example, applications of quantum chemical methods to the study of interstellar molecules (Wilson 1980d)). The results of all calculations at one level of approximation constitute what Pople (1973) has termed a ‘theoretical model chemistry’. The effectiveness of any model may be evaluated by comparing some of its details with data derived from experiment when it is available. If the results of such a comparison are favourable, the model acquires some predictive credibility and can be used to study molecules for which no experimental data is available.

To qualify as a satisfactory theoretical model chemistry, a method should ideally satisfy a number of conditions and requirements:

(i) It should be ‘conventional’. The method should have been applied to a wide range of molecular systems and its strengths and weaknesses should be well documented. Thus when the method is applied to a molecular system, for which experimental information is not available, the likely accuracy of the results can be inferred and empirical corrections to the calculated results can sometimes be made.

(ii) It should provide well-defined results for the energy of any electronic state for any arrangement of fixed nuclei. The method will then afford a set of continuous potential energy curves or surfaces.

(iii) It should be such that the amount of computation is not too large and should not increase too rapidly with the number of electrons in the system. If a method requires a great deal of computer time in order to obtain the desired accuracy, it would be both expensive and time-consuming to test the method by applying it to a wide variety of molecules.

(iv) The method should enable meaningful comparisons of molecules of different sizes to be made. Pople (1973) refers to this property as size-consistency. The application of a technique to two well-separated molecules should yield results which are additive for the energy and for other properties. The widely-used method of configuration mixing, limited to single- and double-excitations with respect to some single-determinantal, or indeed multideterminantal, reference function does not satisfy this requirement.

(v) The calculated electronic energy should ideally be an upper bound or lower bound to the true energy.

(vi) The model should be the simplest possible which will afford the required results.

1.4 Potential energy curves and surfaces

Solution of the electronic Schrödinger equation yields the electronic energy of a molecule for the particular electronic state under consideration for a given fixed position of the nuclei. The electronic energy thus depends parametrically on the nuclear coordinates. These electronic energy values, when regarded as a function of the nuclear configuration, give the potential energy curve or surface on which the nuclei move.

For a diatomic molecule, let us denote the potential energy curve by U(R), where R is the internuclear separation. This function is included in the Schrödinger equation for the nuclear wave function, which takes the form

µ is the reduced mass

in this equation. Equation (1.4.1) may be separated by using spherical polar coordinates and putting

where Yl,m(θ, ϕ) is a spherical harmonic. This leads to a radial wave equation of the form

where J = 0, 1,… is the rotational quantum number. The nuclear wave function is thus separated into a vibrational function and a rotational function.

The separation of the vibrational and rotational motion in polyatomic molecules requires the construction of the Eckart vectors and Eckart frames (Eckart, 1935). We shall not consider this problem further here but refer the reader to recent reviews (Louck and Galbraith 1976; Sutcliffe 1980).

For a diatomic molecule, if the potential energy curve has been calculated then, in principle, the vibrational and rotational levels may be obtained by solution of the nuclear wave equation. The terms neglected in the separation of the electronic and nuclear parts of the hamiltonian give rise to small diagonal corrections which may be added to the potential energy function. The nuclear Schrödinger equation for diatomic molecules can be solved directly by numerical integration, following the approach first described by Cooley (1961) and Cashion (1963). The major drawback of this direct method for solving the nuclear wave equation is that it is necessary to employ a very large number of points on the curve in order to obtain the required accuracy. Some method of interpolation, therefore, has to be used.

A simpler method is to solve the vibrational equation

and treat the term J(J + 1)/2R² as a small perturbation. The vibrational equation can be solved either by the Cooley–Cashion method or by expanding χ(R) in some set of functions and invoking the variation theorem.

In the vast majority of calculations of vibrational and rotational levels in diatomic molecules, approximate solutions of the electronic Schrödinger equation are obtained in tabular form at a relatively small number of internuclear distances. In order to consider vibrational and rotational effects, or to calculate spectroscopic constants or examine molecular dynamics, it is necessary to interpolate between the calculated values. Power series expansions provide a reasonably general means of achieving this. We note, however, that Padé approximants may also be of some use in this respect (Jordan, Kinsey, and Silbey 1974; Jordan 1975).

There are four factors which can influence the expansion coefficients and hence the calculated vibrational frequencies, rotational constants and equilibrium geometries in a power series expansion for the potential energy function:

   (i)   The accuracy of the calculated points;

   (ii)  The order of the polynomial fitted to the calculated points;

  (iii) The range, number and distribution of the points to which the power series is fitted;

   (iv) The expansion variable involved.

In connection with the first two factors, it has been shown (Meyer and Rosmus 1975; see also Wilson 1978a) that there is some advantage in fitting different orders of polynomial to energies obtained by means of an independent electron model, such as the matrix Hartree–Fock method, and also to the correlation corrections. Correlation energies are often not known as accurately as the reference energies and should, therefore, be interpolated by a lower-order polynomial. The third factor is, of course, limited by the total number of nuclear configurations for which electronic structure calculations are performed. It is often useful to test calculations by omitting one or two points from the fitting procedure and observing the effect on the expansion coefficients. With regard to point (iv), there appear to be three choices of expansion variable which can be usefully employed in the power series

The coefficients A0 and ai are to be determined by a least-squares fitting procedure. The Dunham parameter (Dunham 1932; Sandeman 1940) is

where r is the nuclear separation and re is its equilibrium value. Using this parameter a power series is obtained having the radius of convergence

The modified Dunham parameters (Fougere and Nesbet 1966; Simons, Parr, and Finlan 1973; Beckel 1976; Wilson 1978a)

and

are often used. The radius of convergence of the respective power series is given in parenthesis. If the expansion coefficients of the three series obtained by employing ρ1, ρ2, and ρ3 are denoted by A0, a1, a2,…; B0, b1, b2,…; C0, c1, c2…, respectively, then they may be related by the equations

and

Of course, in using the expansion parameters ρ1, ρ2, and ρ3, it has been assumed that re is known. This is not, in general, the case and therefore, an iterative procedure has to be adopted starting from some initial guess for re.

In 1932, Dunham calculated the energy levels of a rotating vibrator using die Wentzel–Kramers–Brillouin method. Using the expansion parameter ρ1 Dunham obtained the energy level formula

where the first subscript under Y refers to the power of the vibrational quantum number and the second that of the rotational quantum number. The Ys are usually referred to as Dunham coefficients. The first fifteen Dunham coefficient are displayed in Table 1.1. The Dunham coefficients are not exactly equal to the spectral constants obtained by means of perturbation theory (see, for example, Herzberg 1950). The connection between the Dunham coefficients and the band spectrum constants is as follows:

Table 1.1

The Dunham coefficients

1.5 Experimental observables

The ultimate test of the theoretical apparatus and computational methods described in this monograph must be made through comparisons with experimentally observed properties of molecules. Often, however, a theoretical technique is assessed not by comparison with experiment but by comparing with results obtained by means of other theoretical approaches. This is because, although the ultimate aim of theory is to reproduce and rationalize experimental data, one can often learn more about a particular model by comparison with theoretically well-defined quantities than by comparison with experiment. For example, it is not possible to determine the Hartree–Fock limit of the energy or any other expectation value from experiment, neither can the correlation energy be measured. When testing a new basis set for self-consistent-field calculations or a new method for performing calculations of electron correlation energies, it is most valuable to compare the resulting energies with the Hartree–Fock energies or with the results of a full configuration interaction calculation in a limited basis. Full configuration mixing within a limited basis set is an exact solution of the Schrödinger equation within that basis set and thus the extent to which a particular model can reproduce that result is an indication of its accuracy, even though in the comparison the basis set truncation error is extremely large.

When the results of a theoretical study are compared with experiment this must be done with care. Comparisons are very often made with quantities which are derived from experimental data rather than with the experimental observations themselves. For example, after calculating a set of points on the potential energy