Hamiltonian, variational principle, Hartree and Hartree-Fock method

The occurs only if is equivalent to , but otherwise, . This theorem can be easily proven (see for example: Levine, 1983; Slater, 1968, or Szabo & Ostlund, 1989). What is more important however, that this theorem provides us with the prescription how to find the good wave function: try to go as low with the expectation value of energy, as you can (you cannot go lower than the true value). And the lower you go, the better you are. The word of caution here... The principle is true only if we are applying a true hamiltonian in the expression for expectation energy. Of course, if your hamiltonian does not represent a physical system, you can get whatever you wish, included!!!

As said before, we can rarely solve the Schrödinger equation exactly. And we have to use approximations.

The first successful attempt to derive approximate wave functions for atoms was devised by Hartree (1928). In this approach the many-electron wave function is approximated by the product of one-electron functions for each of the N electrons:

In this equation, are the positional coordinates and a spin coordinate for the i-th electron. For example, in cartesians , though to solve this equation for atoms, the spherical coordinates are more useful , where can adopt only 2 values: / (spin up, or or ) or / (spin down, , or ). The individual one-electron functions are called orbitals, (or spinorbitals) and describe each electron in the atom (or molecule). Is this a reasonable approximation? Not really...

1.

It assumes that electrons in the atom can be described independently, i.e., their movements do not depend upon each other and their interaction is not pairwise, but each electron interacts with some averaged field of other electrons. This is a neat idea, the problem is that it is not true. Electrons have to avoid each another (correlate their movements), since they repel each other being of the same charge.

2.

The function does not have a proper symmetry for interchanging particle indices for fermions. It was discovered long time ago that the many electron wave function has to be antisymmetric to the exchange of neighboring indices, i.e., change sign:

e.g., .

where

• is the operator of kinetic energy of nuclei,
• represents the kinetic energy of electrons,
• is the interaction energy of nuclei (Coulombic repulsion),
• is the external potential (in this case, the electrostatic potential coming form nuclei, with which electrons interact),
• denotes electrostatic repulsion between electrons.

Since nuclei are much heavier than electrons (a proton is 1836 times heavier than an electron), they have much larger intertia than electrons, and we can consider that electrons will adjust their positions instantly whenever nuclei move. Prof. Rod Bartlett once used an analogy of flies around the wedding cakes - you move the cakes, and the flies move with them. Without much error, we can therefore separate the movement of electrons and nuclei, and assume that the movement of electrons depends on positions of nuclei in a parametric way. This is the contents of the Born-Oppenheimer (1927) approximation. It allows us to use the electronic hamiltonian to study electrons, and add the components of energy coming from nuclei (i.e., their kinetic energy, , and internuclear repulsion energy, ) at the end of calculations.

The form of these operators will be given below. Atomic units are used here, i.e., mass of electron ; ; length is expressed in bohrs (1 bohr = 0.529177249Å; it is the radius of the first Bohr orbit in hydrogen atom); energy is in hartrees (1 hartree is 2 rydbergs, and the ground state energy of hydrogen atom is -1 rydberg, or 0.5 hartree, 1 hartree = 627.51 kcal/mol = 2625.5 kJ/mol); 1 unit of charge is a charge of a proton, the gaussian electrostatic system is used (the unit of permittivity is and it does not appear in the Coulomb law for vacuum).

Here, the is the charge of an -th nucleus (atomic number), is the distance between electron i and the nucleus , is the total number of nuclei in the molecule (but it is obviously equal to 1 for an atom), is the distance between electron i and j. Note, that and can be expressed in cartesian coordinates:

(note that nuclear coordinates for each of the nuclei: , , are constants/parameters rather than variables in the electronic hamiltonian ),

We can rewrite again the electronic hamiltionian:

where operator

and only depends on coordinates of of a given i-th electron. In the product function from equation (17), would only affect the function for the i-th electron. Unfortunately, there is still this which depends on pairs of electrons, and we cannot separate variables in the the Schrödinger equation, (a second order differential equation). So, Hartree found an approximation, in which electron does not interact with other electrons one by one, but with the averaged density of electrons. If we assume for the moment that we know the one-electron functions , for each electron (they make the total wave function in equation (17)) we can calculate densities corresponding to each electron using equation (12):

And the total density of the electrons will be a sum of individual electron densities:

However, the k-th electron does not interact with the whole density , since it is itself a part of it. Electron cannot interact with itself! It is not a human being!!! This would be a self-interaction. So, if we want to find a correct density with which the k-th electron interacts (let us denote it by ), we need to subtract its own density from :

Now, let us assume that we know some approximate functions for the orbitals. And we want to calculate the energy of interation of point charge (our electron) located at position , with other electrons represented as a smeared electron density. Out point charge is -e, which in atomic units is -1, and the electron density is also negative, so we get a positive contribution to energy:

Now, if we make such an approximation, we can write:

(it is not entirely true, as we will see later, since we doubly count interaction, but let us leave it at that for a moment), and our consists of sums of one-electron operators:

and the many electron Schrödinger equation can be solved as N independent one electron equations:

The is the energy of the i-th electron. In practice, we start with some approximate orbitals (e.g., from hydrogen atom). We need them, since the depends on them. We solve all N equations and obtain the N new 's. The new 's are different from the old 's, and we believe that they are better. And now we have better approximations for orbitals, and use the new 's as a starting point and repeat the process. At some point, 's do not change from iteration to iteration, and we obtained the self-consistent field orbitals. From these orbitals we can form a many electron wave function , and then calculate the total energy E of the ground state. Note, that the total energy is not equal to the sum of orbital energies . We calculate the expectation value of energy using the accurate hamiltonian from equation (20). When we solved equation for and we included the Coulomb interactions between electron: (1, 2), (1, 3), (1, 4), etc. When we solved second equation, we included interactions: (2, 1), (2, 3), (2, 4), etc. But the interaction (1, 2) is the same as (2, 1), i.e., adding the energies we would count interactions twice. For this reason, the correct total energy can be represented as:

where 's represent Coulomb interaction of electron i and j. They are called Coulomb integrals and are defined as:

The Hartree approximation works well for atoms. The one-electron functions are quite good, and allow us to produce an approximate many-electron function for the whole atom. But the form of the function adopted by Hartree was basically wrong. It was known at that time that interchanging the electron labels in the wave functions for the system of electrons has to change its sign. This is not something which can be derived. It is simply a law of nature. And it is important, since without this property, the function will not correctly describe the system. Electrons (fermions) have this propery, and we should use it. In the system of fermions, no two particles can be described by the same one particle function. So, few years later, Fock (1930), and independently Slater(1930) proposed a fix to the Hartree method. They also used one-electron functions, but the total wave function for the system was not a simple product of orbitals, but an antisymmetrized sum of all the products which can be obtained by interchanging electron labels. It is conveniently represented as a determinant (called Slater determinant):

Let us examine some properties of this function taking a 2-electron case:

If we change electron labels and we get

that is: . Moreover, if we assume that two electrons are described by the same spinorbital: we get:

i.e., such wave function whould be zero everywhere, and hence, the probablility of finding such electrons is zero. So, Slater determinant satisfies the Pauli exclusion principle that each electron has to be described by a different wave function. It was originally formulated by Pauli (1925) for electrons in atoms, that no two electrons can have the same values of all four quantum numbers: n, l, , and . The single determinant wave function is much better than the Hartree product function as it naturally includes some basic fermion characteristics.

However, it complicates equations compared to Hartree method and introduces a new term, electron exchange. The method of finding the best single determinant wave function for the system is called Hartree-Fock method.

The expectation value of total energy for the single determinant wave function is given by:

where:

is an element of one-electron operator which was defined by equation (25), the is the Coulomb integral defined by equation (34), and is the new term, called exchange integral, which is defined by

Note, that is similar in form to the but the functions and were exchanged. Also, electrons i and j have to be of the same spin for to be nonzero, due to othogonality of their spin parts. It does not have a simple physical interpretation like (i.e., purely electrostatic interaction of two charge densities for electron i and j). The exchange integral comes as a result of the determinantal form of which sums all possible products of permutations of electrons among orbitals. and are equal only when i=j. This is very important, since there is no contribution to the total energy coming from electron self-interaction. It can be shown that 's are larger (or equal to) 's and they are positive numbers.

There are essentially two approaches to HF method: purely numerical, and the one in which orbitals are represented as combinations of basis functions. The numerical HF was used at some point to derive orbitals for many-electron atoms. There are also some numerical HF programs for molecules. However, these methods are quite intensive. It is much more popular to represents orbitals as a linear expansion into a set of some basis functions :

and optimizing coefficients rather than basis functions.

The derivation of specific equations for Hartree-Fock method will not be given here, but can be found in many books, (see, e.g.: McWeeny & Sutcliffe, 1969; Parr, 1963; Pilar, 1968; Slater, 1968; Szabo & Ostlung, 1989) and landmark papers of Roothaan (1951, 1960). In the SFC LCAO MO (Self Consistent Field - Linear Combination of Atomic Orbitals, Molecular Orbitals) method, we are looking for the coefficients which minimize the energy.

They are derived using varational principle, i.e. the goal here is to find a single determinant function as represented by equation (36) which corresponds to the lowest possible value of energy. It is done by the variational calculus. The condition of minimum of energy is:

Another condition is that orbitals have to be orthonormal, i.e.:

where is called a Kronecker delta, and is equal to 1 only if i=j, but is zero otherwise.

The problem of minimization of a function with some subsidiary conditions is done efficiently with the Lagrange's method of undetermined multipliers. In this method, the constraints are converted into expressions whose value is zero when the constraint is met. Such expressions are then multiplied by an undetermined constant and added (or subtracted, according to taste) to the minimized function. The resulting sum is then minimized:

As a result, one gets Hartree-Fock eigenequations.

where the Fock operator is defined as:

In these equations, the fact that operators act on certain coordinates is stressed by writing explicitly their dependence on coordinates of electron 1 (it could be any electron number, since it does not matter how we number them). The was defined in equation (25). The Coulomb operator,

and the exchange operator

are defined by the result of their action on a function.

If we now introduce basis, i.e., replace 's with their expansions into basis functions according to equation (43), the eigenproblems become a set of algebraic equations. Substituting equation (43) into (47) one gets:

and multiplying on the left by and integrating over one gets:

Integrals are further denoted as:

which is an element of a Fock matrix , and

which is an element of a overlap integrals matrix . We can obtain n (i.e., number of basis functions) of such equations, and it is convenient to write them in a matrix form as:

The problem now is to find such marix which diagonalizes , which is a standard procedure in linear algebra. First step is to find a matrix which diagonalizes overlap integrals , which is a step ortogonalizing the basis set (i.e., the basis set is converted to linear combinations of original basis functions, which are mutually orthogonal).

Applying this matrix to equation (55) and rearranging it we get

where and . The efficient routines for matrix diagonalization (i.e., obtaining in this case) are widely available.

While they seem simple, the problem is that elements of depend on orbitals , i.e., one can solve them only through iterative process as in Hartree's method. Moreover, due to Coulomb and exchange operators, the Fock matrix elements involve a massive number of two-electron integrals of a type:

To escape these difficulties and complexities, people tried for many years to decribe electron systems using density, rather than wave function. In the HF approach the two-electron integrals dominate the computational effort. Moreover, the HF is an approximation, as it does not account for dynamic correlation due to the rigid form of single determinant wave function. To solve the HF equations, the assumption has to be made that electrons interact with the averaged potential coming from other electrons, while in fact, the interactions between electrons are pairwise. In reality, electrons correlate their movements trying to avoid each other, so there is least amount of electrostatic repulsion. To account for dynamic correlation, one has to go to correlated methods, which use multideterminant wave functions, and these scale as fifth, or even greater powers with the size of a system. While HF method is quite successful for geometries, it fails miserably to describe bond breaking or forming (for review of correlated ab initio methods see: Bartlett & Stanton, 1994).

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