Last Updated 11 March 1998

Tight-Binding Overlap Parameters

In the paper Applications of a new tight-binding total energy method for transition and noble metals: Elastic constants, vacancies, and surfaces of monatomic metals [Phys. Rev. B 54, 4519-30 (1996)], we assumed the overlap matrix elements had the form

Sll'm(R) = (a + b R + c R2) exp(-d2 R) f(r) , (1)

where a, b, c, d were fitting parameters and f(r) is a cutoff function. The parameters l and l' indicate the angular momentum of the two wave functions, and m indicates the behavior of the matrix element under rotation. Obviously we must have some additional conditions on S. For example, since S is supposed to represent overlapping normalized wave functions, we must have

|Sll'm(R)| < 1 (2)

for all R > 0.

If two atoms involved in the overlap are identical, then (1) is obviously too general. In particular, if l'=l, we have

Sllm(0) = 1 , (3)

while if l'<>l we must have

Sll'm(0) = 0 , (4)

failure to respect (3) and (4) will lead to situations where the overall overlap matrix becomes unphysical, i.e., has negative eigenvalues. In this case the tight-binding procedure fails because we cannot solve the generalized eigenvalue problem. In general this will not cause difficulties when we stay close to the structures included in the fit, but it may cause problems when we go far beyond the fitted structures.

To help alleviate this difficulty, in Tight-Binding Hamiltonians for Carbon and Silicon[MRS Proceedings] we introduced a new form for the parametrization of the overlap matrix elements when both atoms are of the same type. When l' = l, this the new overlap has the form

Sllm(R) = (1 + a R + b R2 + c R3) exp(-d2 R) f(r) , (5)
while for l'<>l we use

Sll'm(R) = (a R + b R2 + c R3) exp(-d2 R) f(r) . (6)
Obviously (5) and (6) automatically satisfy the condtions (3) and (4). They do not necessarily satisfy (2), however, and we are considering other forms for the parametrization.

The parameter files for the new forms, (5) and (6), have the same structure as the old parameter files. To allow programs such as static to distinguish between the two forms we use the new header information on the parameter files to designate the new overlap parametrization as style "00001". The old parametrization is "00000".

For older versions of static we used the following convention:

Since the first parameter in the file, lambda, is squared before it is used to calculate the fictitious charge density, we adopt the convention that lambda>0 means to use the overlap parametrization (1), while lambda<0 means to use the overlap parametrizations (5) and (6) for like atoms. Note that we still use (1) for unlike atoms, since these wave functions need not obey (3) and (4).

We are still looking for a better way to control the overlap matrix elements, so the form of the overlap parameters may change again in the future. (In this case there will be a new version number.)

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