Current functional theory for multi-electron configuration

Abstract

The density functional theory (DFT) formalism is reformulated into a framework of currents so as to give the energy a parameter dependent behaviour, e.g., time. This “current” method is aimed at describing the transition of electrons from one orbital to another and especially from the ground state to an excited state and extended to the relativistic region in order to include magnetic fields which is relevant especially for heavy metallic compounds. The formalism leads to a set of coupled first order partial differential equations to describe the time evolution of atoms and molecules. The application of the method to ZnO and H₂O to calculate the occupation probabilities of the orbitals lead to the results that compare favorably with those obtained from DFT. Furthermore, evolution equations for electrons in both atoms and molecules can be derived. Applications to specific examples of small molecules (being metallo-oxides and water) are mentioned at the end.

Appendices

Appendix 1: An expression for an exchange-correlation energy

Let us briefly mention an earlier derivation of an exchange-correlation expressed as a difference, Q = V − V _H between the total potential V and the Hartree potential, V _H , and with the help of a new central potential term, also used by Talman and Shadwick, see Ref. [13], as well as in Eqs. 11–16. We may then write:

$$ Q=\sum\limits_i \sum\limits_{j\ne i} \int\limits {\rm d} {\bf {r}}^{\prime3} \int\limits {\rm d} {\bf {r}}^{\prime\prime3} \frac{\psi_i ({\bf r})^{{\ast}} \psi_k ({\bf r}^{\prime})}{E_i - E_j} 2 \times\sum\limits_k\frac{ \psi_j (r)^{{\ast}} \psi_k ({\bf r^{\prime}}) \psi_k ({\bf r^{\prime\prime}})^{{\ast}} \psi_i ({\bf r^{\prime\prime}})}{\mid {\bf r^{\prime}} - {\bf r^{\prime\prime}} \mid} $$

(104)

which can be simplified using the same restriction on the indices of Coulomb matrix elements, as in Ref. [2]:

$$ Q=C \sum\limits_i \sum\limits_{j\ne i} \frac{\psi_i ({\bf r})^{{\ast}} \psi_k ({\bf r}^{\prime})}{E_i - E_j} \int\limits {\rm d}r^{\prime} \frac{({\bf j}^{L}_{jk}({\bf r^{\prime}}) {\bf j}^{L}_{kl}({\bf r^{\prime}}))}{(E_k - E_i)(E_j - E_k)} $$

(105)

where $C=\hbar 8\pi$. It is quite remarkable that one can obtain such relatively simple expression for the exchange-correlation. This is a contribution between different orbitals or quantum states i, j. In the case of E _i = E _j this term vanishes and we are left with the zero’th order Born term. For details see Ref. [2].

Appendix 2: Transition charge density

For the sake of consistency of the paper this appendix brings a derivation of the expression, following the prescription of Ref. [13] for the transition current densities that are used in Sects. 3 and 4 of the present paper. We start, as in Ref. [13] from an expression for the one-particle transition charge density:

$$ \rho(r)_{ki} = e \psi(r)_k^{{\ast}} \psi(r)_i $$

(106)

where ψ(r) _k , ψ(r) _i are the one-particle energy eigen-states and where the numbers i, k stand for the radial, angular momentum,..., etc. The perturbative matrix element for the residual Coulomb interaction between two electrons situated at r and r′ is then expressed by the integral:

$$ I_C = \int\limits {\rm d}r^3 \int\limits {\rm d}r^{\prime 3} \frac{\rho(r)_{ki} \rho(r^{\prime})_{jl}}{| {r - r^{\prime } }|} $$

(107)

and applying the time-dependent Schrödinger equation for ψ the integral can be written as

$$ I_C = - \int\limits {\rm d}r^3 \int\limits {\rm d}r^{\prime3} \frac{1}{| {r - r^{\prime}}|}\frac{\partial \rho(r)_{ki}}{\partial t} \frac{\partial \rho(r^{\prime})_{jl}}{\partial t} (E_{ki} E_{jl})^{-1} (\hbar )^2 $$

(108)

where E _ij = E _i  − E _j . If one again applies the continuity equation of transition charge densities one can write the integral I _C :

$$I_C = - \int\limits {\rm d}r^3 \int\limits {\rm d}r^{\prime3} \frac{1}{| {r - r^{\prime}|}} {\rm div}_{r} {\bf j(r)}_{ki} {\rm div}_{r^{\prime}} {\bf j(r^{\prime})}_{jl} (E_{ki} E_{jl})^{-1} (\hbar )^2$$

(109)

with j _ij being the transition current:

$$ {\bf j}_{ij} = a (\psi_i^{{\ast}} \nabla \psi_j - \psi_j \nabla \psi_i^{{\ast}}) \equiv {\bf j}^{\alpha}{\bf j}^{\beta} $$

(110)

where $a = \frac{e \hbar}{2m i}$ and where α, β correspond to the quantum numbers of the interchange of the states i,j. One should remark that, fortunately, if the denominator E _ij is zero then the numerator is also zero.

We are now in position to derive the very important expressions for the current that are used in Sect. 3. These expressions have been derived with the powerful tool of vector algebra where the vectors, in this case the transition current densities, are split into longitudinal parts, j ^L, and transverse parts, j ^T as seen below:

$$ {\bf j(r)}^L = - {\rm grad}_{r} \int\limits {\rm d}{\bf r}^{\prime3} {\rm div}_{r^{\prime}}{\bf j(r^{\prime})}\frac{1}{(4 \pi \mid {\bf r}-{\bf r^{\prime}} \mid )} $$

(111)

which in terms of densities is:

$$ {\bf j(r)}_{ij}^L = \int\limits {\rm d}r^{\prime3} {\rm grad}_{r^{\prime}}{\bf \rho(r^{\prime})}_{ij} \frac{iE_{ij}}{(4 \pi \mid {\bf r}-{\bf r}^{\prime} \mid \hbar )} $$

(112)

and

$$ {\bf j(r)}^T = {\rm curl}_{r^{\prime}} \int\limits {\rm d}{\bf r}^{\prime3} {\rm curl}_{r^{\prime}}{\bf j(r^{\prime})}\frac{1}{(4 \pi \mid {\bf r}-{\bf r^{\prime}} \mid )} $$

(113)

which altogether can be written as ${\bf j} = {\bf j}^L + {\bf j}^{T}$.

With these expressions inserted in Eq. 146 and subsequently performing a partial integrations one finally obtain:

$$ I_C = \frac{\hbar^2 4 \pi}{E_{ij} E_{kl}} \int\limits {\rm d}{\bf r}^3 {\bf j(r)}_{ki}^L {\bf j(r)}_{jl}^L \equiv \frac{\hbar^2 4 \pi I^L} {E_{ij} E_{kl}} $$

(114)

which therefore transform the sixth-dimensional integral at the start of the appendix to a much simpler three-dimensional integral of the scalar product of the two transition current densities. There are similar expressions valid for the transverse component. However, only the longitudinal ones are used for the calculation of the Coulomb matrix elements.

Appendix 3: Relativistic electrons

Another interesting problem is connected with relativistic effects. The kinetic energies of electrons in atoms and molecules are only in the neighbourhood of the very heaviest nuclei of such an order of magnitude that relativistic effects come into play in a direct way. However, magnetic interactions can always be considered as relativistic effects, and a proper treatment as that may be, not only consistent, but also the simplest way, as shown below.

The electric current density in the Dirac theory of relativistic electrons is given as the 3-vector part of a 4-vector:

$$ s_{\nu=1,2,3,4}=iec\varvec{\bar{\Uppsi}}\gamma_{\nu}\varvec{\Uppsi} $$

(115)

$$ s_{k=1,2,3}=j_k;...s_4=ic\rho $$

(116)

The 4-vector s _ν can be decomposed

$$ s_{\nu}=s^C_{\nu}+s^P_{\nu} $$

(117)

so that the space part of s ^C, the conduction current is of the same form as the non-relativistic expression for j _k . The remaining part is the polarization current s ^P _k . The Dirac State vector is here

$$ \varvec{\Uppsi}=\sum\limits_{\nu}\psi_{\nu}{\bf e_{\nu}} $$

(118)

and the conjugate

$$ \varvec{\bar {\Uppsi}}=\varvec{\Uppsi}^\dagger\alpha_0= \sum\limits_{\nu}{\bar \psi}_{\nu}{\bf e}^T_{\nu} $$

(119)

where ${\bf e}^T_{\nu}$ stands for the transposed of ${\bf e}_{\nu}$.

Here the ψ are functions of x, y, z, t and the e are orthogonal unit vectors in spin space. The Dirac equation is now

$$ \begin{aligned} &\left[{\bf \alpha}_0mc+\sum\limits_k{\bf \alpha}_k\left(\frac{\hbar}{i} \frac{\partial}{\partial{x_k}}-\frac{e}{c}A_k\right)+ e\phi\right] {\boldsymbol \Uppsi}\\ &=\left[{\bf \alpha}_{0}mc{\boldsymbol \Uppsi}+{\alpha}{\cdot}\left(\frac{\hbar}{i}\nabla{\boldsymbol \Uppsi} -\frac{e}{c}{\bf A} {\boldsymbol \Uppsi} \right)+e\phi{\boldsymbol \Uppsi}\right] =-\frac{\hbar}{ic} \frac{\partial{\boldsymbol \Uppsi}}{\partial t} \end{aligned} $$

(120)

Here ${\bf \alpha}$ is a vector with components α _k and α₀ = ρ₀, the familiar Dirac matrices. The continuity equation in this relativistic case becomes

$$ \frac{\partial(\varvec{\Uppsi}^\dagger \varvec{\Uppsi})}{\partial{t}} =-c \;{\rm div}(\varvec{\Uppsi}^\dagger\alpha\varvec{\Uppsi}) $$

(121)

Any field in three-dimensions may be split in a longitudinal and a transverse part

$$ {\bf j}={\bf j}^{\rm long}+{\bf j}^{\rm tran} $$

(122)

$$ {\bf j}^{\rm long} =-\nabla_r\int\limits {{\rm d}^3r^{\prime}}\frac{\rm div_{r^{\prime}}{\bf j}(r^{\prime})}{4\pi |{\bf r-r^{\prime}}|} $$

(123)

Let the transition current field be given by its relativistic form in spherical coordinates

$$ {\bf j}^1_{ki}=iec{\bar {\boldsymbol \Uppsi}_{k}}{\bf \gamma} {\boldsymbol \Uppsi}_i $$

(124)

with the spinors

$$ {\boldsymbol \Uppsi}_{i,J_i,l_i,M_i} ={\cal F}(r,t)_i {\cal Y}_{lm}\chi_{\mu}\left(lm\frac{1} {2}\mu|JM\right). $$

(125)

In the presence of a magnetic field, the one electron current density takes the form

$$ {\bf j}_{ki}^{\bf A}=\frac{\hbar}{2mi}\left({\boldsymbol \Uppsi}_k^{\ast}{\nabla}{\boldsymbol \Uppsi}_i-{\boldsymbol \Uppsi}_i{\nabla}{\boldsymbol \Uppsi}_k^{\ast} -\frac{2ie}{c\hbar}{\bf A \boldsymbol \Uppsi}_k^{\ast}{\boldsymbol \Uppsi_{\bf i}}\right) $$

(126)

or in spherical form

$$ {\bf j}^A=C_1{\bf j}^1_{JlM}-C_2({\bf A}\rho)_{JlM} $$

(127)

where the C’s can be derived from equations of the earlier subsections e.g., (45)–(47) and $({\bf A}\rho)_{JlM}$ is given by the definition of vector spherical harmonics in the expression

$$ {\bf A}\rho={\bf A} \sum\limits_{mq}Y_{lm}{\bf e}_q(lm1q|JM) $$

(128)

Remember that A is a vector, ρ a scalar. The longitudinal part of these vectors is now found in the same way as in Appendix 2 and in Ref. [13]. In order to treat the magnetic interactions between electrons and between nuclei as well as between nuclei and electrons, we shall look at the transverse transition current densities. At the same time we shall introduce a new form of these current densities, appropriate for the interactions with magnetic fields:

$$ {\bf s}_{ki}={\bf j}_{ki}-\frac{e^2}{c} {\bf A} \boldsymbol{\Uppsi}^{\ast}_k\boldsymbol{\Uppsi}_i $$

(129)

$$ {\bf s}^T={\rm curl}_r\int\limits {\rm d}^3{\bf r}^{\prime}\frac{{\rm curl}_{r^{\prime}}({\bf s}({\bf r}^{\prime})-\frac{e^2}{c} {\bf A} \rho({\bf r}^{\prime}))}{4\pi{|{\bf r}-{\bf r}^{\prime}|}} $$

(130)

with

$$ \rho({\bf r}^{\prime})=\Uppsi^{\ast}_k({\bf r}^{\prime})\Uppsi_i({\bf r}^{\prime}) $$

(131)

The new expression for the current density gives the same continuity expression as the previous one. The curl A = H gives the connection to the magnetic field (which could originate from other electrons or nuclei, static or in transitions in general (j − h)). The magnetic interaction energy [ki, jh] can again be written by means of the integral of a scalar product of two transition current densities, ${\bf j}^T_{ki}$ and ${\bf j}^T_{jl}$. Note, however that the two proportionality constants in the electric and magnetic energies, expressed by means of currents, are different. To show the effect of the method of giving the matrix elements by means of currents,we shall start from the final result and work backwards:

$$ I_M=\int\limits {\rm d}^3r({\bf j}({\bf r})^T_{ki}.{\bf j}({\bf r})^T_{jh})C_{ki,jh} $$

(132)

where

$$ {\bf j}({\bf r})^T={\rm curl}_r\int\limits {\rm d}^3{\bf r}^{\prime} \frac{{\rm curl}_{{\bf r}^{\prime}}{\bf j}({\bf r}^{\prime})}{4\pi|{\bf r}-{\bf r}^{\prime}|} $$

(133)

(Note, however, that the constant C in the magnetic matrix elements is different from the corresponding constant in the electric matrix element).

Note also, that the current in the magnetic matrix element must include the electromagnetic potential A, and that the transverse current densities have the same form as the source terms of electric multi-pole radiation (and vice versa with magnetic multi-pole radiation).

The occurrence of curl j in the last equation above and elsewhere may seem inconvenient, since this expression is not part of the common formulae, derived from Maxwell’s or Schrödinger’s equations. Introducing

$$ {\bf j^T=j-j^L} $$

(134)

this formal problem is avoided.

Appendix 4: Transverse and longitudinal parts of F _{α, J, J} ′

This last appendix will bring a few essential formulas for the vector fields that are used in the present article.

The transverse and longitudinal vector fields are, in the spherical vector representation, defined by the following equations:

$$ {\bf T}_{J, J, M} = F(r)_{J, J} {\bf Y}_{J, J, M}, \ {{\rm with}\ J\ {\rm integer}\ (J > 0)} $$

(135)

and

$$ {\bf L}_{J, M} = F(r)^{L}_{J+} {\bf Y}_{J, J+1, M} + F(r)^{L}_{J-} {\bf Y}_{J, J-1, M}, \; {\rm with}\; L\; {\rm integer}\; (l = J \pm 1) $$

(136)

We can furthermore specify the transverse vector-field, like the longitudinal field above, by using the expression of the curl of the field T so that:

$$ {\bf T}_{J, M} = \sqrt{\frac{J}{2J+1}}({\rm d}/{\rm d}r -J/r)\Upphi^T_J {\bf Y}_{J, J+1, M} + \sqrt{\frac{J+1} {2J+1}} ({\rm d}/{\rm d}r + (J+1)/r)\Upphi^T_J {\bf Y}_{J, J-1, M} $$

(137)

where $\phi^T_J = F_{J,J}(r)$ and, concerning the longitudinal field, $F(r)^L_{J+},F(r)^L_{J-}$ are given by:

$$ F(r)^L_{J+} = - \sqrt{\frac{J}{2J+1}}({\rm d}/dr -J/r)\Upphi^L_J $$

(138)

and

$$ F(r)^L_{J+} = -\sqrt{\frac{J}{2J+1}}({\rm d}/{\rm d}r +(J+1)/r)\Upphi^L_J $$

(139)

We can now bring the connection of these vector fields to our electron functions u, v:

$$ u(r)_J = \sqrt{\frac{J}{2J+1}}\Upphi^T_J - \sqrt{\frac{J+1} {2J+1}}\Upphi^L_J $$

(140)

and

$$ v(r)_J = \sqrt{\frac{J}{2J+1}}\Upphi^L_J + \sqrt{\frac{J+1} {2J+1}}\Upphi^T_J $$

(141)

with the transition current being obtained by equating the coefficient of ${\bf Y}_{J, l, M}$ above:

$$ {\bf j}_{\alpha} = \sum\limits_{J, l=J\pm 1, M} F_{\alpha, J, l}{\bf Y}_{J, l, M} $$

(142)

We shall finish the appendix by just giving the first terms for F _J,M:

$$ F_{1,0}= -(4\pi 3)^{-1/2} r e_{-r}2e^{-2r}={\rm exp}[-3r](-3-r)/\sqrt{4\pi } $$

(143)

and

$$ F_{1,2}= -(4\pi 3/2)^{-1/2} r e_{-r}2e^{-2r} - (4\pi)^{-1/2}e^{-2r}re^{-r}(2/3)^{1/2}= {\rm exp}[-3r]r \sqrt{2/3}/ \sqrt{4\pi} $$

(144)

and therefore

$$ ({\rm d}/{\rm d}r - 1/r)u_1 = F_{1,2} = {\rm exp}[-3r]r \sqrt{2/3}/\sqrt{4\pi} $$

(145)

and similar for v ₁.

Appendix 5: Gaussian expansions and numerical calculations

In the following, we want to be able to compare results with standard DFT calculations using Gaussian functions. Hence, we shall, for the purpose of comparisons, introduce Gaussian expansions of the wavefunctions and similarly of the density ρ and its derivative that are also taken as Gaussian like. It is thus natural to start from the current expression:

$$ j_{ki}^L = \nabla_{r} \int\limits {\rm d}r^{\prime3} \rho (r^{\prime})_{ki}^A iE_{ki}/((r_ )4\pi\hbar )Y_{J,J-1,M} $$

(146)

and each Gaussian term now gets a radial form:

$$ w(r) =r^{-A} \int\limits_0^r {\rm d}r^{\prime} r^{\prime A} \rho(r^{\prime}) C^{\prime} $$

(147)

The argument from Ref. [13] that concerns the finiteness of the integrals above, independent of the sign of A, is still valid. Actually, extending the upper integration limit to infinity renders these Gaussian integrals manageable, such that:

$$ \int\limits_0^{\inf} {\rm e}^{-\lambda x^2}x^k = 1/2 \lambda^{-(k+1)/m}/\Upgamma((k+1/m) $$

(148)

showing that no singularities are occurring in this formalism. The integrals, which we shall use, are of the form:

$$ Q(r) = -\sum\limits_i \int\limits {\rm d}r^{\prime} \int\limits {\rm d}r^{\prime\prime} \psi_i (r)^{{\ast}} G_i (r,r^{\prime}) V_e (r^{\prime},r^{\prime\prime}) \psi_i (r^{\prime\prime}) $$

(149)

which can be simplified, using the procedures of Eqs. 103 and 106, to:

$$ r^{-A} \int\limits_0^r {\rm d}r^{\prime} (r^{\prime A}) C \int\limits_0^{r^{\prime}} {\rm d}r^{\prime\prime} {\rm e}^{[-r^{\prime\prime2}]} $$

(150)

where A = J + 1 corresponds to the vector spherical harmonics to be Y _J,J−1,M but, in the case of A =  −J, becomes Y _{J,J + 1,M}. Considering just one term (diagonal) in a Gaussian expansion of an electron density (an ordinary, non-transition density) ρ(r) _kk , we can use the above formula to get:

$$ \frac{j(r)_{kk}^L}{E_{kk}}= \int\limits {\rm d}r^{\prime3} \nabla^{\prime}_r \rho (r^{\prime})_{kk} i/ 84\pi \hbar $$

(151)

where

$$ \begin{aligned} \nabla_r[\rho(r)] &= \nabla_r (F(r)Y_{JM}) \\ = & -\sqrt{(J+1)/(2J+1)}({\rm d}/{\rm d}r - J/r)F(r)Y_{J,J+1,M} \\ & + \sqrt{J/(2J+1)}({\rm d}/{\rm d}r-J/r)F(r)Y_{J,J-1,M} \\ \end{aligned} $$

(152)

The two terms correspond to A =  −J and A = J + 1, respectively. Terms with negative A can be eliminated in the radial integral, where the F-functions now are of the Gaussian type. This can be seen by using a recursive procedure, repeated some times:

$$ i^n erfc(z) = -z/ni^{n-1} erfc(z) + 1/(2n)i^{n-2} erfc(z) $$

(153)

or

$$ 2(n+1)(n+2)i^{n+2} erfc(z) = (2n +1+2z^2)erfc(z) - 1/2i^{n-3} erfc(z) $$

(154)

where erfc is the error function.