## Extended Koopmans theorem matrix elements.

Keyword: EKT

Description: This is to calculate the ionization energy and electron affinity using the extended Koopmans' theorem (EKT). The EKT module compute the following matrix element,

(1) One-body reduced density matrix. $\rho_{ij}=\langle \Psi^N |c_i^\dagger c_j|\Psi^N\rangle$, where $\Psi^N$ is the wave function of the N-electron system, and $c_i$ is the annihilation operator in second quantization. This matrix elements of $\rho$ is evaluated as while \$\phi_i\$ is the orbitals by which the density matrix is represented with.

(2) Electron ionization matrix. $V_{ij}^v = \langle \Psi^{N}|c_i^\dagger c_j H -c_i^\dagger H c_j |\Psi^{N}\rangle$ where $H_n$ is the hamiltonian corresponds to the n-th particle (including kinetic energy, external potential energy, and coloumb energy from other electrons).

(3) Electron affinity matrix. $V_{ij}^c = \langle \psi^N|c_i [\hat H, c_j^\dagger]|\psi^N\rangle$, which is evaluated as where $\hat h_0$ is the kinetic term, and $v(\vec r_i, \vec r_j)$ is the Coulomb interacting term. The ionization spectrum is obtained by solving the following generalized eigenvalue problem, Suppose the number of electron in the system is N. The inonization spectrum corresponds to the lowest N eigenvalues of the equation. The electron affinity spectrum is obtained by The electron affinity spectrum corresponds to the eigenvalues start from N+1. The EKT should converge as the number of states included increases. However, the stochastic uncertainty also increases with more states. It is often useful to try a few different cutoffs on the number of states in order to converge the results. Therefore, in solving the generalized eigenvalue problem, we might include only {1 2 3 ... Ns'}, we increase Ns' before the stochastic error goes up.

The output of the EKT can be read in using functions from utils/read_dm.py.

Required keywords

Keyword Type Description
ORBITALS/CORBITALS section Input for a set of orbitals. When ORBITALS section is used, a real-valued wave function is constructed. When a CORBITALS section is used, the orbitals will be complex-valued.

Optional keywords

Keyword Type Default Description
STATES section All the orbitals in the ORBITALS section. A list of the orbitals on which you'd like to evaluate the density matrices.<

## Manybody polarization operator

Keyword: MANYBODY_POL

Description: $z_i=\left\langle \exp(-i{\mathbf k_i} \cdot {\mathbf X}) \right\rangle$, where ${\mathbf X} = \sum_j {\mathbf r_j}$ and ${\mathbf k_i}$ is a reciprocal vector of the supercell. The magnitude of $z_i$ is related to the localization length in the direction of the reciprocal lattice vector. Its phase gives the center of charge in the unit cell.

Required keywords

None

Optional keywords

None

## Pairwise region fluctuations

Keyword: REGION_FLUCTUATION

Description: We define the number operator on as site i for spin $\sigma$ as $n_{i,\sigma}= \sum_{j\in \sigma} \Theta(r_j)$, where j is summed over the electrons in the spin $\sigma$. This object calculates the joint probability distribution function between all pairs of sites and spins, $\rho( n_{i,\sigma},n_{j,\sigma} )$.

The output can be read in by the Python functions in utils/read_numberfluct.py.

Required keywords

None

Optional keywords

Keyword Type Default Description
MAXN integer 20 The maximum number of electrons you expect to see on any atom

## Static structure factor

Keyword: SK

Description: Electronic static structure factor given by $S({\bf k})=\frac1N\,\langle\Psi|\hat\rho_{\bf k} \hat\rho_{-\bf k}|\Psi\rangle$ where $\hat\rho_{\mathbf{k}}$ is a Fourier component of the electron density. The quantity is evaluated either on a full 3D grid or on a 1D grid along a fixed direction, see keyword DIRECTION.

The structure factor $S({\bf k})$ can be used to evaluate finite-size corrections to the Ewald energy, see Chiesa et al. Phys. Rev. Lett. 97, 076404 (2006). Note that $S({\bf k})$ as calculated contains besides the continuous electronic component also sharp Bragg peaks, since electronic wave function follows crystal symmetries. These are the same peaks that X-ray diffraction uses to determine crystal structure.

Required keywords

None

Optional keywords

Keyword Type Default Description
NGRID integer 5 Number of grid points in one direction. Structure factor is evaluated on NGRID points if DIRECTION is given and on NGRID$^3$ points otherwise.
GVEC section reciprocal lattice vectors Basis vectors ${\bf g}_i$ defining the grid. The section contains a 3x3 matrix of floats.
DIRECTION integer not set Selects one of the three basis vectors ${\bf g}_i$, along which $S({\bf k})$ is evaluated.

## Two-body density matrix on a basis

Keyword: TBDM_BASIS

Description: This object evaluates the one and two-body reduced density matrix on a basis. The basis can be any set of functions that can be expressed in a \ref MO_matrixDoc object. Usually this will be a set of orbitals from a mean-field calculation. The ORBITALS section can often be copied from the .slater file with no changes. The output contains the states section and the density matrices. One thing to keep in mind is that the density matrices are always labeled from 0 to nmo, where nmo is the number of orbitals in the STATES section.

Required keywords

Keyword Type Description
ORBITALS/CORBITALS section Input for a set of orbitals. When ORBITALS section is used, a real-valued wave function is constructed. When a CORBITALS section is used, the orbitals will be complex-valued.

Optional keywords

Keyword Type Default Description
STATES section All the orbitals in the ORBITALS section. A list of the orbitals on which you'd like to evaluate the density matrices.
MODE string Evaluate all two-particle density matrix elements. Possible values are OBDM and TBDM_DIAGONAL.