Calculation of DOS and Greenís function*

 

The definitions of the one-electron Greenís function for interacting system is

 

†††† (1.1)

 

where

††††††††††††††††††††††††††††††††††††††††††††††††††††††††

Here ďLĒ denotes left eigenvector to distinguish it from the right ďRĒ eigenvector,

is the chemical potential, are the eigenvalues, is the self-energy, is the Hamiltonian

and is the overlap matrix which takes into account the non-orthogonality of the basis set

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ,

and is defined as follows

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† .

 

The orthonormality condition is

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ,

or in the matrix form

.

 

Now one can easy see by inspection that the Hamiltonian can be written in the form:

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ,††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††

where is a diagonal matrix of eigenvalues. Using the above expression for the Hamiltonian one can easily prove the above equation for the Greenís function:

††††††††††††††††††††††††††††††††††

In direct space, formula for the Greenís function is

††††††††††††††††††††††† ,

or

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ,

where the wave functions are

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ,

which also satisfy to the following equations

†††††††††††

The Equation for the GF in direct space is just the Fouriesr transformation of ††taking into account the basis set equation:

.

Density can be found from the Greenís function directly

††††††††††††††††††††††††††

Integration over the space gives the interacting density matrix which should be defined with the overlap matrix

†††††††††††††††††††††††††††††††††††††††† ,

in order to preserve the sum rule related to the total number of electrons

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† .

 

Note, that as soon as we assume that our basis set is orthonormal, we restore ordinary definitions for all the relevant quantities discussed above.

 

The total density of states (TOS) is defined as

 

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† .

Unfortunately, the definition

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ,

has a problem since it does not have proper symmetry transformation. By this we mean that

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ,

where are Wigner matrices andis any group operation. Assuming by the rotation of the entire Brillouin zone

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† ,

we see that we cannot prove this equality. We can easily prove this for objects like

A possible way to resolve it can be found if the overlap matrix can be represented as

††††††††††††††

†††††† (1.2)

where ††is the structure constants and are the overlap matices of Hankel and Bessel functions .

This representation is true in the Atomic Sphere Approximation (ASA) only. If such factorization possible, we can define the non-interacting density of states as

††††††††††††††

†††††††††††††† (1.3)

 

where by we denoted convolutions

†††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††† .

It is clear that each term obeys the required transformation property, and so does the resulting

Next we consider the basis orthonormalization procedure. One can show that the overlap matrix can be presented in the form

††††††††††††††

†††† (1.4)

where we made the following definitions

.

Here h and j are matrices which are calculated by minimization such that Eq. (1.4) is best satisfied.

Let us denote transformation which we use to bring the system to the orthonormal basis

††††††††††††††

††††††† (1.5)

In the almost orthonormal basis the Greenís function reads as

††††††††††††††

†††††††††††† (1.6)

 

where

†††††††††† (1.7)

Having GF g we can compute the density of states as

††††††††††††††

†† (1.8)


Using GF and the self-energy we can also compute the Weis GF using the following relation:

. (1.9)


The impurity levels can be obtained from the frequency expansion of the impurity and local GFs:



,(1.10)

where matrix contains is the Hartree-Fock values of the self-energy,isthe double counting term. As we see to compute the impurity levels we need to know the following averages (which are matrixes): .
If we are interested only in diagonal elements of the impurity matrix then the expression (1.10) is simplified to


,(1.11)

where values of are provided by the k-sum driver/program.



 

* Most of the equations in the text related to LMTO ASA method can be found in

O. K. Andersen, Linear methods in band theory,Phys. Rev. B 12, 3060 (1975).

 

 

Last updated @ Sept 6 2004

 

 

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