Perturbation approach to constrained electron transfer in density functional theory

Abstract

As an alternative to the variational constrained density functional theory a simpler perturbative approach based on the Taylor series expansion, around the ground state of the system, of the density matrix and the electronic density, as functions of the Lagrange multiplier, introduced to impose the constraint in the variational procedure, is developed and implemented in deMon2k. The first and second order corrections introduced are evaluated with the information contained in the ground state calculation, through the use of auxiliary density perturbation theory, while the value of the Lagrange multiplier at which the series is evaluated is determined from the amount of charge transferred. The energy differences between the ground and the constrained states obtained with the first and second order perturbative terms lie, in general, close to the variationally determined values. In addition, it is found that the results are practically independent of the exchange–correlation energy functional used and show rather small variations with respect to the basis set.

Appendix: Implementation in deMon2k of the first and second order corrections to the density matrix

In order to describe the main aspects of the first and second order perturbation corrections to density matrix and to the electronic density, we will make use of a matrix language, since its implementation is done in a developer version of deMon2k [27] that is based on a Gaussian basis set.

Thus, to obtain the first order perturbed density, we use ADPT [22,23,24,25, 48, 49]. In this approximation the first order density is obtained through the McWeeny self-consistent perturbation theory [50,51,52],

$$P_{{\mu \nu }}^{{\left( \lambda \right)}} = 2\,\sum\limits_{i}^{{occ}} {\sum\limits_{a}^{{uno}} {\frac{{K_{{ia}}^{{\left( \lambda \right)}} }}{{\varepsilon _{i} - \varepsilon _{a} }}\,\left( {c_{{\mu i}} \,c_{{\nu a}} + c_{{\mu a}} \,c_{{\nu i}} } \right)} } ,$$

(30)

where the sum over $i$ corresponds to the occupied molecular orbitals, while the sum over $a$ corresponds to the unoccupied ones. In ADPT the first order Kohn–Sham matrix is obtained with the following expression,

$$K_{{ia}}^{{\left( \lambda \right)}} = \left\langle {i|(w_{{Ac}} ({\mathbf{r}}) - w_{{Do}} ({\mathbf{r}}))|a} \right\rangle + \sum\limits_{{\overline{k} }} {\left\langle {ia\left\| {\overline{k} } \right.} \right\rangle \,\left( {x_{{\overline{k} }}^{{\left( \lambda \right)}} + z_{{\overline{k} }}^{{\left( \lambda \right)}} } \right)} ,$$

(31)

where

$$z_{{\overline{k} }}^{{\left( \lambda \right)}} = \sum\limits_{{\overline{l} }} {\sum\limits_{{\overline{m} }} {\,G_{{\overline{k} \overline{l} }}^{{ - 1}} f_{{\overline{l} \overline{m} }} \,x_{{\overline{m} }}^{{\left( \lambda \right)}} } } ,$$

(32)

$x_{{\overline{k} }}^{{\left( \lambda \right)}}$ is the $k{\text{th}}$ element of the first order auxiliary density, $f_{{\overline{l} \overline{m} }}$ represents the integral of the second variational derivative of the exchange–correlation functional with two auxiliary functions $l$ and $m$.

For the particular case of this new perturbation, we solved the ADPT equations as made in previous work [22,23,24,25, 48]

$$\left( {{\mathbf{G}} - 4{\mathbf{A}} - 4{\mathbf{A F}}} \right)\,{\mathbf{x}}^{{\left( \lambda \right)}} = 4{\mathbf{b}}^{{\left( \lambda \right)}} ,$$

(33)

where ${\mathbf{F}} = {\mathbf{G}}^{{ - 1}} {\mathbf{f}}$, ${\mathbf{G}}^{{ - 1}}$ is the inverse of the Coulomb matrix ${\mathbf{G}}$, and the elements of the Coulomb coupling matrix can be expressed as

$$A_{{\overline{k} \overline{l} }} = \sum\limits_{i}^{{occ}} {\sum\limits_{a}^{{uno}} {\frac{{\left\langle {\overline{k} \left\| {ia} \right.} \right\rangle \left\langle {ia\left\| {\overline{l} } \right.} \right\rangle }}{{\varepsilon _{i} - \varepsilon _{a} }}} } .$$

(34)

The elements of the perturbation vector ${\mathbf{b}}^{{\left( \lambda \right)}}$ contain the explicit form of the perturbation, that is

$$b_{{\overline{l} }}^{{\left( \lambda \right)}} = \sum\limits_{i}^{{occ}} {\sum\limits_{a}^{{uno}} {\frac{{\left\langle {i|(w_{{Ac}} ({\mathbf{r}}) - w_{{Do}} ({\mathbf{r}}))|a} \right\rangle }}{{\varepsilon _{i} - \varepsilon _{a} }}\left\langle {ia\left\| {\overline{l} } \right.} \right\rangle } } .$$

(35)

Once the first order auxiliary density coefficients have been obtained by solving Eq. (33), the first order Kohn–Sham matrix, Eq. (31), can be evaluated, and with this one, the first order density matrix can be determined.

Now, in the framework of ADPT the explicit form of the second order perturbed density matrix is given by [25, 49]

$$P_{{\mu \nu }}^{{\left( {\lambda \lambda } \right)}} = 2\,P_{{\mu \nu }}^{{\left( {\lambda ,\lambda } \right)}} + 2\,\sum\limits_{i}^{{occ}} {\sum\limits_{a}^{{uno}} {\frac{{K_{{ia}}^{{\left( {\lambda \lambda } \right)}} }}{{\varepsilon _{i} - \varepsilon _{a} }}\,\left( {c_{{\mu i}} \,c_{{\nu a}} + c_{{\mu a}} \,c_{{\nu i}} } \right)} }$$

(36)

This expression was derived to obtain third derivatives in the energy, required for the calculation of first hyperpolarizabilities. However, it was not implemented in its explicit form, because the Wigner’s 2n + 1 rule allowed to calculate third derivatives of the energy in terms of first order densities. Here we illustrate how the implementation of the explicit form of Eq. (36) was performed to obtain second order densities. The first term in the right-hand side of this expression depends on the following first order densities,

$${\mathbf{P}}_{{}}^{{\left( {\lambda ,\lambda } \right)}} = {\mathbf{P}}_{{oo}}^{{\left( {\lambda ,\lambda } \right)}} + {\mathbf{P}}_{{uu}}^{{\left( {\lambda ,\lambda } \right)}} + {\mathbf{P}}_{{ou}}^{{\left( {\lambda ,\lambda } \right)}} + {\mathbf{P}}_{{uo}}^{{\left( {\lambda ,\lambda } \right)}} .$$

(37)

Now, in order to simplify the explicit form of these matrices, one can introduce the transformation matrix,

$$W_{{\mu \nu }}^{{(\lambda )}} = \,\sum\limits_{i}^{{occ}} {\sum\limits_{a}^{{uno}} {\frac{{K_{{ia}}^{{\left( \lambda \right)}} }}{{\varepsilon _{i} - \varepsilon _{a} }}\,c_{{\mu i}} \,c_{{\nu a}} } } ,$$

(38)

and its transpose, as

$$W_{{\mu \nu }}^{{(\lambda )\dag }} = \,\sum\limits_{i}^{{occ}} {\sum\limits_{a}^{{uno}} {\frac{{K_{{ia}}^{{\left( \lambda \right)}} }}{{\varepsilon _{i} - \varepsilon _{a} }}\,c_{{\mu a}} \,c_{{\nu i}} } } .$$

(39)

The occupied-occupied block of Eq. (37) is expressed as follows in terms of this transformation matrix,

$${\mathbf{P}}_{{oo}}^{{\left( {\lambda ,\lambda } \right)}} = - {\mathbf{W}}^{{(\lambda )}} \,{\mathbf{S}}\,{\mathbf{W}}^{{(\lambda )\dag }} - {\mathbf{W}}^{{(\lambda )}} \,{\mathbf{S}}\,{\mathbf{W}}^{{(\lambda )\dag }} .$$

(40)

where ${\mathbf{S}}$ corresponds to the overlap matrix. The unoccupied-unoccupied block is calculated as

$${\mathbf{P}}_{{uu}}^{{\left( {\lambda ,\lambda } \right)}} = {\mathbf{W}}^{{(\lambda )\dag }} \,{\mathbf{S}}\,{\mathbf{W}}^{{(\lambda )}} + {\mathbf{W}}^{{(\lambda )\dag }} \,{\mathbf{S}}\,{\mathbf{W}}^{{(\lambda )}} ,$$

(41)

while the occupied-unoccupied block is given by

$${\mathbf{P}}_{{ou}}^{{(\lambda ,\lambda )}} = \, - \sum\limits_{i}^{{occ}} {\sum\limits_{a}^{{uno}} {\frac{{\,2\,\,\Omega _{{ia}}^{{\left( {\lambda ,\lambda } \right)}} \,}}{{\varepsilon _{i} - \varepsilon _{a} }}\,{\mathbf{c}}_{i}^{{}} \,{\mathbf{c}}_{a}^{\dag } } } .$$

(42)

Here, ${\mathbf{c}}_{i}^{{}}$ is a vector whose elements are the coefficients of the molecular orbital $i$, equivalently for ${\mathbf{c}}_{a}^{{}}$, and

$$\Omega _{{ia}}^{{\left( {\lambda ,\lambda } \right)}} = {\mathbf{c}}_{i}^{\dag } \left( {{\mathbf{K}}^{{(\lambda )}} \,{\mathbf{W}}^{{(\lambda )}} \,{\mathbf{S}} - {\mathbf{S}}\,{\mathbf{W}}^{{(\lambda )}} \,{\mathbf{K}}^{{(\lambda )}} } \right){\mathbf{c}}_{a}^{{}} ,$$

(43)

and the unoccupied-occupied block of Eq. (37) is given by

$${\mathbf{P}}_{{uo}}^{{(\lambda ,\lambda )}} = {\mathbf{P}}_{{ou}}^{{(\lambda ,\lambda )\dag }} .$$

(44)

Now, the second term of Eq. (36) depends on the second order Kohn–Sham matrix, which can be expressed as

$$K_{{ia}}^{{\left( {\lambda \lambda } \right)}} = \sum\limits_{{\overline{k} }} {\left\langle {ia\left\| {\overline{k} } \right.} \right\rangle \,\left( {x_{{\overline{k} }}^{{\left( {\lambda ,\lambda } \right)}} + \sum\limits_{{\overline{l} }} {\left( {{\mathbf{1}} + {\mathbf{G}}^{{ - 1}} {\mathbf{f}}} \right)_{{\overline{k} \overline{l} }} x_{{\overline{l} }}^{{\left( {\lambda \lambda } \right)}} } } \right)} ,$$

(45)

where ${\mathbf{x}}^{{\left( {\lambda ,\lambda } \right)}}$ depends on first order densities,

$$x_{{\overline{k} }}^{{\left( {\lambda ,\lambda } \right)}} = \sum\limits_{{\overline{l} }} {\sum\limits_{{\overline{m} }} {\sum\limits_{{\overline{n} }} {G_{{\overline{k} \overline{l} }}^{{ - 1}} g_{{\overline{l} \overline{m} \overline{n} }} x_{{\overline{m} }}^{{\left( \lambda \right)}} x_{{\overline{n} }}^{{\left( \lambda \right)}} } \,} } ,$$

(46)

and $g_{{\overline{l} \overline{m} \overline{n} }}$ is the integral of the third variational derivative of the exchange–correlation functional with three auxiliary functions $l$, $m$ and $n$.

In the calculation of first hyperpolarizabilities, this third variational derivative was integrated with three first order densities. The second order auxiliary coefficients, ${\mathbf{x}}^{{\left( {\lambda \lambda } \right)}}$, are obtained by solving the system of equations,

$$\left( {{\mathbf{G}} - 4{\mathbf{A}} - 4{\mathbf{A G}}^{{ - 1}} {\mathbf{f}}} \right)\,{\mathbf{x}}^{{\left( {\lambda \lambda } \right)}} = {\mathbf{b}}^{{\left( {\lambda ,\lambda } \right)}}$$

(47)

with

$$b_{{\overline{k} }}^{{\left( {\lambda ,\lambda } \right)}} = 2\,\sum\limits_{\mu } {\sum\limits_{\nu } {P_{{\mu \nu }}^{{\left( {\lambda ,\lambda } \right)}} \left\langle {\mu \nu \left\| {\overline{k} } \right.} \right\rangle } } + 4\,\sum\limits_{{\overline{l} }} {A_{{\overline{k} \overline{l} }} x_{{\overline{l} }}^{{\left( {\lambda ,\lambda } \right)}} }$$

(48)

Thus, after solving Eq. (47) one can obtain the explicit elements of the second order Kohn–Sham matrix, Eq. (45), and once the second order density matrix is determined, one can incorporate the first and second order corrections to the density.

It is important to note that the present derivation is limited to the local density (LDA) and generalized gradient (GGA) approximations to the exchange–correlation energy functionals, since in the case of hybrid functionals with a fraction of exact exchange, one would have to include additional terms for the calculation of the first and second order perturbation corrections that have not been considered in the previous equations.

A relevant aspect of the present approach is that it may be readily implemented in any code that includes the calculation of polarizabilities and hyperpolarizabilities, through simply replacing the electric field by the term associated to the constrained charge transfer, $(w_{{Ac}} ({\mathbf{r}}) - w_{{Do}} ({\mathbf{r}}))$, which is what we did to arrive at Eqs. (31) and (35).