Constrained–random-phase–approximation formalism

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The constrained random-phase approximation (CRPA) is a method that allows to calculate the effective interaction parameter U, J and J' for model Hamiltonians. The main idea is to neglect screening effects of specific target states in the screened Coulomb interaction W of the GW method. The resulting partially screened Coulomb interaction is usually evaluated in a localized basis that spans the target space and is described by the model Hamiltonian. Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher level theory, such as dynamical mean field theory (DMFT).


Model Hamiltonians

A model Hamiltonian describes a small subset of electrons around the chemical potential and has, in second quantization, following form

Here, are site and spin indices, respectively and the symbol indicates summation over nearest neighbors. The hopping matrix elements describe the hopping of electrons (of same spin) between site and , while the effective Coulomb matrix elements describe the interaction of electrons between sites.

Wannier basis and target space

To use model Hamiltonians successfully a localized basis set is chosen. In most applications this basis set consists of Wannier states that are connected with the Bloch functions of band at k-point with spin via

Usually, the basis set is localized such that the interaction between periodic images can be neglected. Hence, in practice one works with the Wannier function in the unit cell at and writes instead:

In practice one selects only a subset of Bloch functions, typically around the chemical potential (i.e.


Parameter definitions

Mind: The calculation of the hopping matrix depends on the theory that is used to solve the effective model.

For instance, in DFT+DMFT (often termed LDA+DMFT) one calculates the hopping matrix from the Kohn-Sham energies, while in GW+DMFT the GW quasi-particle energies are used. If denotes these one-electron energies and is the corresponding Fermi energy, the hopping matrix elements are calculated with following formula

Similarly, the Coulomb matrix elements are evaluated from the Bloch representation of the effective Coulomb kernel via

Mind: The effective Coulomb kernel is frequency dependent and thus the effective interaction in the model as well.

In most applications, however, one considers the static limit .

In practice one often, simplifies the model Hamiltonian further and works with the Hubbard-Kanamori parameters:[1]

Here specifies the number of Wannier functions in the basis set.

Effective Coulomb kernel in constrained random-phase approximation

In analogy to the screened Coulomb kernel in GW, the effective coulomb kernel is calculated as

In contrast to the GW method, however, the polarizability contains all RPA screening effects, except those from the target space. These effects can be obtained with the target Bloch states:

Using Green's functions of the target space

the polarizability of the target space reads

After a Fourier transform to reciprocal space and imaginary frequency axis one ends up with

describing the propagation within the target space.

=

Related tags and articles

ALGO, NTARGET_STATES, NCRPA_BANDS LDISENTANGLE LWEIGHTED NUM_WANN WANNIER90_WIN ENCUTGW VCUTOFF

References