Keyword POPULATION

This keyword selects a population analysis.
Options:
MULLIKEN / LOEWDIN / BADER / NBO / BECKE / HIRSHFELD / VORONOI

	MULLIKEN		A Mulliken population analysis is performed. This is the default.
	LOEWDIN		A Löwdin population analysis is performed.
	BADER		A Bader population analysis is performed.
	NBO		Create input file for an NBO analysis.
	BECKE		Real space is partitioned into overlapping (fuzzy) Becke cells.
	HIRSHFELD		Real space is partitioned into overlapping Hirshfeld cells.
	VORONOI		Real space is partitioned into non-overlapping Voronoi cells.

SCF / ITERATIVE / DEFORMATION / FULL

	SCF		The SCF density is used for the Hirshfeld weights. Only meaningful for the HIRSHFELD option.
	ITERATIVE		The Hirshfeld weights are refined iteratively. Only meaningful for the HIRSHFELD option.
	FRACTIONAL		Same as the ITERATIVE option but separates $\alpha$ and $\beta$ spin densities. Only meaningful for the HIRSHFELD option.
	DEFORMATION		The deformation density is analyzed. Only meaningful for the BECKE, HIRSHFELD and VORONOI options.
	FULL		More information is printed.

Description:
For all population analyses, deMon2k calculates atomic charges and, in the case of open-shell systems, atomic spin populations. From the orbital-based population analysis of Mulliken [262] or Löwdin [263], the bond order [264] or valence matrix [267] is calculated as well (closed-shell systems only) with the option FULL. In the case of the Bader population analysis [38], the critical points of the electron density and the molecular graph are also calculated. These quantities can be visualized with VU [41] using the deMon.pie file. The option BADER together with the option FULL prints out extra information about the dipole and quadrupole moments of atoms and non-nuclear attractors as well as information about their kinetic energies. For other options related to the Bader population analysis see (4.10.2). The input file for the NBO population analysis is created for use with the NBO program [268]. The options MULLIKEN, LOEWDIN, BADER, NBO, BECKE, HIRSHFELD and VORONOI are mutually exclusive.

With the options BECKE, HIRSHFELD or VORONOI a topological atom-in-molecule (AIM) analysis of the electronic density is performed. The electronic density, $\rho(\mbox{\boldmath$r$})$ is integrated numerically on a fixed grid of points. By selecting the BASIS or AUXIS option of the TOPOLOGY keyword (see 4.10.2), the user triggers the analysis of the Kohn-Sham density (BASIS) or of the auxiliary density (AUXIS). By default the Kohn-Sham density is used for all topological analyses. The quality of the integration grid is also set with the TOPOLOGY keyword using the options COARSE, MEDIUM, FINE or REFERENCE. The DEFORMATION option of the POPULATION keyword triggers the analysis of the deformation density, $\Delta \rho(\mbox{\boldmath$r$})$, defined as:

$$\Delta \rho({\bf r}) = \rho^{SCF}({\bf r}) - \rho^{Pro}({\bf r})%$$ (37)

In Eq. (4.28) $\rho^{SCF}({\bf r})$ is the relaxed electronic density produced by the SCF and $\rho^{Pro}({\bf r})$ is the promolecular density obtained by the superposition of spherically averaged atomic densities. According to the TOPOLOGY options BASIS or AUXIS these are either the corresponding Kohn-Sham (default) or auxiliary densities, respectively. Having defined the electronic density to analyze, the number of electrons, $N_A$, is calculated for each atom $A$ according to:

$$N_A = \sum_{i}^{\rm Grid} \rho({\bf r}_i) \, \omega_q({\bf r}_i) \, \omega_A({\bf r}_i)%$$ (38)

The summation runs over all the grid points. The quadrature weights $\omega_q$ are calculated according to the Euler-McLaurin (radial) and Lebedev (angular) quadrature schemes. The atomic weights $\omega_A$ determine how the electron density is distributed over the atoms. Their mathematical definition is different for each topological population scheme. In particular, the following atomic weight definitions are implemented for the VORONOI, BECKE and HIRSHFELD options.

Voronoi:

$$\omega_A({\bf r}_i) = \left \{ \begin{array}{ll} 1 & \fora... ...t close to atom }A \\ 0 & \mbox{else} \end{array} \right.%$$ (39)

Becke:

$$\omega_A({\bf r}_i) = { P_A({\bf r}_i) \over { \displaystyle{\sum_B P_A({\bf r}_i)}}}%$$ (40)

Hirshfeld:

$$\omega_A({\bf r}_i) = { {\tilde{\rho}}_A^{ref}({\bf r}) \over { \displaystyle{\sum_B {\tilde{\rho}}_B^{ref}({\bf r})}}}%$$ (41)

The cell function $P_A({\bf r}_i)$ of the BECKE weights is defined by a "soft" step function in terms of the elliptic coordinates $\mu_{AB}$ defined as [269]:

$$\mu_{AB} = {{\bf r}_A - {\bf r}_B \over {{\bf R}_{AB}}}%$$ (42)

The geometrical construction of these coordinates is depicted in Figure 13.

Figure 13: Elliptic coordinate definition used for the cell function of BECKE atomic weights.

As can be seen from Eq. (4.32) the atomic Hirshfeld weights are always calculated from the auxiliary densities, independent of whether the Kohn-Sham or auxiliary density is partitioned. For the reference densities in Eq. (4.32) 4 different options are available. By default spherically averaged neutral atom densities are used to calculate the atomic Hirshfeld weights. This is the standard Hirshfeld scheme from the literature [265]. With the SCF option of the POPULATION keyword $\rho_A^{ref}$ is calculated from the atomic SCF auxiliary densities. With the ITERATIVE option $\rho_A^{ref}$ of each atom is calculated from a spherically averaged ion with the current charge of the atom (see Eq. 19 of Ref. [266]). Note that this approach is computationally more involved than the standard Hirshfeld scheme because it requires repetitive integration of the electronic density over the grid. The tolerance criteria and the maximum number of iterations are set by the TOPOLOGY keyword options TOL and MAX, respectively. The iterative Hirshfeld scheme produces higher partial charges in absolute value than the standard one and is also applicable to ionic molecules. Moreover the arbitrariness in the definition of the reference densities of the standard Hirshfeld approach is removed with the iterative version. In open-shell molecules either the density or the spin densities can be refined iteratively according to scheme that was just discussed. This selection is triggered by the options ITERATIVE or FRACTIONAL, respectively. Therefore, the input lines,

 POPULATION HIRSHFELD ITERATIVE

and

 POPULATION HIRSHFELD FRACTIONAL

yield the same iterative Hirshfeld charges for closed-shell molecules but different charges for open-shell molecules.