The Hartree self-consistent field method

The Hartree self-consistent field method is an iterative procedure that optimises an approximate wavefunction using the variational principle. Its goal is to estimate the eigenvalues of a modified form of the non-relativistic multi-electron Hamiltonian. This method was developed by the English scientist Douglas Hartree in 1927.

In an earlier article, we showed that the three dimensional Hamiltonian operator for an n-electron atom (excluding spin-orbit and other interactions) is:

where , and .

Our objective now is to solve eigenvalue equations for . The eigenvalue equation of the hydrogen atom has an exact solution (i.e. a solution represented by a formula). However, for an atom with more than one electron, the inter-electronic repulsion term poses a problem, which is a three-body problem that has no exact solution. Therefore, approximation methods are needed.

Firstly, we simplify the Hamiltonian by rewriting it in a system devised by Hartree called “atomic units” (abbreviated to a.u.), where ( is the Bohr radius, where ). For example, if the charge of a particle is 3.4 a.u., it is in S.I. units. Eq1 therefore becomes

with the eigenvalues of also in atomic units.

Secondly, we approximate the form a multi-electron wavefunction takes. As hydrogenic wavefunctions have exact solutions, it is logical to express the multi-electron wavefunction as products of orthonormal one-electron spatial wavefunctions :

Thirdly, we utilise the variational principle, which states that the calculated energy is always greater than or equal to the ground state of any system. This implies that we can calculate an upper bound to of any system by using any trial function . The closer the trial function is to the actual or best wavefunction of the system , the closer E will be to .

Using eq3 and eq2, we have

Simplifying the above equation and minimising E (see this article for proof), we obtain the following set of n ‘one-particle’ equations called the Hartree equations:

where .


How do we interpret eq5?


Eq5 can be perceived as a set of n Schrodinger equations, each consisting of a modified one-electron non-relativistic Hamiltonian. Solving each of the n equations gives a minimised value of corresponding to the i-th electron.

To further explain the modified inter-electronic repulsion term, we refer to eq2. Focussing on electron 1, we have

We regard electrons 2, 3, …, n as being smeared out into a continuous and spherically symmetrical charge distribution through which electron 1 moves. For example, the interaction between electron 1 and electron 2 is

where is the charge density of (i.e. charge per unit volume) of electron 2.

Since , and , we have and hence . So,

The total potential energy between electron 1 and all other electrons is therefore


Since eq5 is the outcome of minimising eq4, subject to the constraints of the orthonormality of one-electron wavefunctions that make up the overall wavefunction of a system, any solution set of from eq5 is associated with a set of , each of which is also obtained from multiplying eq5 by and integrating over all space:

where and .

Summing all of eq6, we have

Since for the Coulomb term (see this article for explanation),

To establish the relation between E and , we rewrite eq4 as

where we have used the fact that is equivalent to (see this article for explanation).

Combining eq7 and eq8,

E is greater than the ground state of the system, except when the solution set is associated with the set of one-electron wavefunctions that best describes the ground state of the system, in which case E is the best estimate of . This best set of and the corresponding set of are found using the Hartree self-consistent field method, which involves the following steps:

  • Make guesses for all single-electron wavefunctions except for one, e.g. use trial functions for except . The corresponding equation to solve is:

Solving the above equation means finding and , which can only be done numerically because of the presence of unknown functions and variables (see this article for an example). The solutions are then the first approximations of and , which leads to a first improvement to eq3.

  • To further improve the wavefunction of eq3, we solve for and using

  • As per the previous step, the same trial functions for are employed together with the computed . The solutions are then the 1st approximation of and , and hence a further improvement to eq3.
  • Next, and derived from the previous two steps are used along with trial functions for to solve for and . The process is repeated until and are solved. The entire cycle is then repeated, i.e. step 1 is repeated using improved wavefunctions of to solve for an even better approximation of , and then followed by step 2 and step 3. This cyclical process stops when all and become invariant, which implies that the best approximation of eq3 for the particular system has been derived.
  • Finally, E is computed with eq9 using the derived and .

As the numerical method of calculating E is usually quite complex, an analytical method is often used. Such a method involves finding analytical expressions for all terms in eq8 using trial one-electron wavefunctions. As per the numerical method, guess values are used for the variables in all trial one-electron wavefunctions except one, e.g. . Eq8 is then minimised to obtain the solutions for E and . The process is repeated until E and all become invariant.

In conclusion, the Hartree self-consistent field method produces relatively accurate results when compared to experiment data for the ground state energy of a small atom like helium. However, it is less accurate for finding E of larger atoms, as it does not take into consideration exchange forces due to spin exchange interactions. A more accurate method, call the Hartree-Fock method is needed for such atoms.


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