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Iterations Towards Self-Consistency

  Just as the LMTO method breaks naturally into an atomic part and a `solid' part, so do the programs. In general the `solid' part requires potential parameters and structure constants as its input, from which it generates bands, energy moments, densities-of-states, etc. The `atomic' part takes moments as its input and calculates potential parameters from it. The atomic part requires very little information beyond the moments and E tex2html_wrap_inline1736 's or boundary conditions to completely specify the electronic structure within an atomic sphere.

An LMTO calculation is self-consistent when the atomic part produces, from moments generated by the solid part, once again the same potential parameters that the solid part used to generate the moments in the first place. The self-consistency works by alternating between the solid part and atomic part, generating moments, then potential parameters, then moments again until the process converges. The program can be started either with the solid part, specifying potential parameters, or with the atomic part, specifying the moments.

Because the method is a linear one, only three functions can carry charge inside a sphere ( tex2html_wrap_inline1812 ) and therefore the properties of a sphere, within the approximations of the linear method are completely determined by the first three moments, the atomic number and the E tex2html_wrap_inline1736 's or boundary conditions at the surface of the sphere. In some sense these numbers are `fundamental' to a sphere; the atomic program will generate a self-consistent potential for a specified set of moments and E tex2html_wrap_inline1736 's and generate potential parameters from it.

From the point of view of the bands, the Hamiltonian is completely specified by the potential parameters (and the structure constants). These are fundamental to the band program; it will generate moments from the eigenvectors of the Hamiltonian. Full self-consistency is achieved when the `input moments' coincide with the `output moments', or equivalently when the input pp's coincide with the output pp's.

The ASA program can start equally as well from either potential parameters or moments, though it is generally easier to start from the moments, if no prior information is available. This is because one can usually begin with a very simple starting guess (choosing the zeroth moment to be the charge of the free atom, the first and second to be zero) that is usually good enough to iterate towards self-consistency.

If potential parameters are available, you may choose to begin directly with a band calculation and need not worry about the moments. If you wish to make a self-consistent calculation, you must also supply the principal quantum numbers for each tex2html_wrap_inline1720 channel in the so-called P number described in the following paragraphs.

To make a sphere self-consistent one needs the moments and also to specify the boundary condition on the wave function at the sphere radius, or the E tex2html_wrap_inline1736 of the wave function tex2html_wrap_inline1710 . For a given potential, there is a unique correspondence between the logarithmic derivative D tex2html_wrap_inline1736 at the sphere radius and E tex2html_wrap_inline1736 , so in principle, it is possible to specify either one.

There is a set of numbers called P (one for each tex2html_wrap_inline1720 channel) that supplies the information about both the principal quantum number and the logarithmic derivative, D. P is defined as: P = .5 - atan(D tex2html_wrap_inline1736 )/ tex2html_wrap_inline1800 + n where n (its integer part) is the principal quantum number; its fractional part varies smoothly from 0 (for the bottom extreme of the band for that principal quantum number) to 1 (the top extreme of the band), and can be thought of as a `continuously variable' principal quantum number. Here is a table of P tex2html_wrap_inline1834 as function of D tex2html_wrap_inline1736 : D tex2html_wrap_inline1736 10 5 1 0 -1 -2 -3 -4 -10 P tex2html_wrap_inline1834 .03 .06 .25 .5 .75 .85 .90 .92 .97

P must be supplied to the atom program. The fractional part of P is automatically supplied by the output of a band calculation (provided the token IDMOD in category CLASS is not 1), but you must supply P (in addition to the moments Q) if you choose to begin with moments. P, together with the moments Q can be input directly through the control file. A word on choices for the fractional part of P: .3 is quite free- electron-like and suitable for free-electron-like tex2html_wrap_inline1720 channels such as Si d electrons, while .8 is quite tight-binding-like and suitable for deep states like Cu d orbitals. If there is no information from the very beginning, .5 is a suitable choice.

In self-consistency cycles, you have a choice, through the parameter IDMOD described below, regarding the related pair of parameters P and E tex2html_wrap_inline1736 . You may let P and E tex2html_wrap_inline1736 float to the center of gravity of the occupied part of the band (most accurate for self-consistent calculations); this is the default. You may also freeze alternatively P or E tex2html_wrap_inline1736 in the self-consistency cycle.

The problem of `ghost' bands can be overcome by using the downfolding procedure. Orbitals are separated in lower, intermediate and higher sets. By switching on the automatic downfolding procedure, this choice is done automatically by the program. Lower waves contribute to the dimension of the Hamiltonian matrix, H, and the overlap matrix, O, and carry charge. Intermediate waves do not contribute to the dimension of H and O, but carry charge. Higher waves are thrown away altogether. If it is not known how to separate the orbitals the automatic downfolding can be switched on before the self-consistency cycle. After a few iterations, the downfolding indices don't change and if it is desired the cycle can be stopped and started again with the proper division into lower, intermediate, and higher set. Calculations using the automatic downfolding should be checked carefully. The token which makes the decomposition is IDXDN.
IDXDN = 0 : automatic downfolding
IDXDN = 1 : low orbitals
IDXDN = 2 : intermediate orbitals
IDXDN = 3 : higher orbitals
More about this below under category CLASS and token IDXDN.

next up previous contents
Next: Accurate band structure away Up: The STUTTGART TB-LMTO-ASA program. Previous: Token LMXRHO= of cast

O. Jepsen
Thu Oct 12 14:48:45 MESZ 2000