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Changes to this chapter were last made on 16 January 2005. |
In this section, a brief overview of the formal foundations of density-functional theory will be given [56]
[58][59][60][2][3]
[Leach,p.127].
The seminal work on DFT appeared in two key papers in 1964 and 1965. In the first paper, it was proved by Hohenberg and Kohn (HK) [56]
that for a N-particle system all of the electronic ground state properties are determined by the
electron density
where the kinetic energy and electron-electron repulsion energy operators are
and v(r) is an arbitrary external
potential that is often
assumed simply to be the electron-nucleus attraction energy, in which case Eq. (6.1) would
be the same as Eq. (5.3). The total energy,
where we have used the fact that
We can repeat the derivation of Eq. (6.7) with the primed and unprimed quantities interchanged to obtain
By adding Eqs. (5.15) and (5.16) we find the following contradiction
Equation (6.11) is justified by the first HK theorem, which established that
The exchange-correlation functional
The functional F[ρ] in Eq. (6.12)
is defined by the constrained-search for the minimum value of
This is a condensed version of the proof originally given by Levy [73][74] and Lieb [75]. It has a slightly broader range of validity than the one first put forth by HK [56]. Those interested in a more thorough discussion should read the reviews by Parr and Wang [2], Kohn [60], and Koch and Holthausen [3].
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One year and three days after Hohenberg and Kohn presented their groundbreaking work setting the
formal foundations of density-functional theory, Kohn and Sham (KS) provided a practical procedure to use DFT
in calculations of electronic structure [58]. The derivation of the Kohn-Sham orbital equations and the
self-consistent field (SCF) energy minimization procedure is similar to the derivation of the
Hartree-Fock equations in Section 5.2. As in the Hartree-Fock case we can express the
N-electron wavefunction in terms of
of orthonormal one-electron spin states,
For the general case of an arbitrary number of spin-orbitals, the kinetic energy functional
The Kohn-Sham orbitals can be written in a form similar to the spin-orbitals defined in Eq. (2.37)
where for the last term in Eq. (6.17) we have used the perhaps too subtle notation that without
a spin label of some sort (i.e., x or σ)
The variational approach to the energy eigenvalue problem is analogous to the derivation of the Hartree-Fock equations. Thus, the stationary condition
leads to the Euler-Lagrange equation
with the constraint being the orthonormality of the Kohn-Sham spin-orbitals
and the Lagrange multipliers,
In Eq. (6.26), we have explicitly shown the spin labels, and one must keep in mind that KS spin-orbitals
themselves are functionals of the density, i.e.,
To evaluate the Euler-Lagrange equation [Eq. (6.24)], we will use
the chain rule for functional derivatives. If, for example, we consider some arbitrary functional
F[y], where
Equation (6.27) will be valid if w(r) is a function of y(r). It can also be obtained from the chain rule by using [Parr, p.250]
If the preceding discussion was a bit confusing, then we can illustrate with an example, which happens
to be the case were interested in. In other words, let
We can use Eq. (6.30) to evaluate the various contributions to the Euler-Lagrange equation from Eqs. (6.11) and (6.12). Starting with the system dependent term containing the external potential, we find
From the system independent functional,
and the contribution from the classical Coulomb energy,
Finally, the contribution from exchange-correlation functional is given by
where we have defined the exchange-correlation potential as
and, as before [i.e., Eq. (5.33)], the contribution from the constraint is
Collecting the terms together gives us the Kohn-Sham orbital equations
where the Kohn-Sham operator is defined as
and the definition of the effective potential is
By comparing Eq. (6.42) with Eqs. (6.11) and (6.12), it can be seen that in a calculation of the total energy we need to correct for the overcounting of the
classical Coulomb energy,
or for closed-shell restricted spin-orbitals we have
which is the form used in our first example program, freeMQM-1.cpp.
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where the function
Equations (6.46) and (6.47) represent the unrestricted Kohn-Sham (UKS) matrix formalism, analogous to the Pople-Nesbet equations. As before, in the case of our Hartree-Fock examples, we will consider closed shell systems and solve the Kohn-Sham equivalent of the Roothaan-Hall equations in the restricted Kohn-Sham (RKS) matrix formalism
and in the RKS formalism it can be written as
where
At this point, we can use Eq. (6.55) together with Green's Theorem in the following form
to calculate the Fock matrix Eq. (6.51), where n is an unit normal vector outward pointing from the surface S. Then, with the following substitutions
we can obtain the following expression [84][86]
where the surface integral contributes nothing when integrated over all space. Equation (6.58) has a form which readily lends itself to numerical integration; it is used by the integration routines to calculate the exchange-correlation (XC) functional contribution to the Fock matrix, which has the variable name FMatrix[I][J] in freeMQM-1.cpp.
To perform a density-functional calculation we will need to make a couple changes
to the procedure used previously for the Hartree-Fock method. The first change that needs to be made is to replace the exchange contribution to the Hartree-Fock equations,
For a Hartree-Fock calculation, the Fock matrix is calculated by the function CalcFockMatrix, which is called from inside the self-consistent loop of the example program freeMQM-1.cpp. In our DFT calculation, CalcFockMatrix still calculates
but the contribution to the Fock matrix from the density-functional,
In general, it can be seen from the form of the function
The routines in functionals.cpp return the values of the function f and the derivatives,
from which we can construct the functional derivative and the exchange-correlation
part of the Fock matrix [Eq. (6.58)]. Further simplifications can be made if we assume
that we are dealing with a closed shell system, in which case the densities of spin-up
are both equal to
As before in the Hartree-Fock problem, the spatial part of the Kohn-Sham orbitals can be written as a sum over our K Gaussian basis functions [Eq. (5.45)]
When applied to density-functional theory (DFT) this is referred to as the restricted Kohn-Sham (RKS) model, as opposed to the unrestricted Kohn-Sham (UKS) formalism, which we might discuss in more detail in a future revision of this document. The file functionals.cpp contains formulas for both the RKS and UKS version of the functionals and their derivatives.
In the closed-shell RKS formalism the gradient of the total electron density is given by
and
To include
and, as before,
Further simplifications can be made for both the helium atom and H_{2}, since they have a single ground-state
orbital occupied by one spin-up electron and one spin-down electron. Therefore the total electron density
is simply
As before [i.e., Eq. (5.66)], the most general form of the primitive Gaussian functions [113]
[114][Szabo, p. 181;Levine,p.462],
In the more general non-spherically symmetric case we will need to evaluate
where the partial derivatives of the Gaussian primitive functions take the more general form
with similar expressions for the partial derivatives with respect to y and z.
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The integration methods used by the functions R_Integrate and xyz_Integrate are summarized in [85]. The function R_Integrate uses the Euler-Maclaurin quadrature formula of the form
respectively. R is referred to as the "atomic radius" [85] and takes the value of 1.0 for the hydrogen atom and 0.5882 for He.
The 3-dimensional quadrature formulas have the following form
where for simplicity, the radial and angular integrations are considered separately.
The angular integrations are performed using the using the method of Gauss quadratures on a sphere
developed by Lebedev (see Refs. [105] to [111]),
and as before
the radial integrations use the Euler-Maclaurin formula [85].
The Euler-Maclaurin-Lebedev (EML) atomic grid is sometimes referred to simply as
The quadrature formulas for the sphere Lebedev and Laikov presented in Ref. [105] are valid for all
polynomials x^{k}y^{l}z^{m},
where
Becke developed a multicenter integration scheme for polyatomic molecules, which we can apply to EML quadratures [88]. In this approach the integral over the polyatomic system is split into individual atomic centered integrals as follows
then the I_{A} can be written as
The problem reduces to finding the weight functions
Following the initial proposal of Becke [88] the weight functions
where
and r_{A} and r_{C} are the distances to nuclei A and C, respectively. R_{AC} is the internuclear distance between nuclei A and C. Becke proposed [88] a smoothly varying analog to the step function
and found that the simplest possible function that could satisfy the necessary constraints was
This does not give a sufficiently rapid cutoff required of the step function; however, three iterations of the function p(μ) has been found to give excellent results [88]
This atomic-centered grid method was one of the first implemented for density-functional theory. Improved descendants are no doubt still used in modern ab initio software packages [Koch,p.107;Treutler and Ahlrichs, 1995][101]. In upcoming sections we will discuss recent developments and alternative methods.
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Much time and effort has been spent trying to obtain exchange-correlation functionals which will give accurate results. Many useful review articles and Internet resources describe some of the most popular functionals. Mark Casida has a relatively exhaustive and up-to-date tabulated listing of most of the density functionals with links to further information. Earlier work includes that of Johnson et al. [86], which provides a discussion of some of the important "2nd generation" functionals, including the Becke (B88) exchange functional [87], the Vosko, Wilkin, and Nusair (VWN) correlation functional [72], and the Lee, Yang and Parr (LYP) correlation functional [96]. This article also includes an appendix with the relevant functional derivatives. The CLRC Quantum Chemistry Group density-functional Repository [50] has a nice summary of many of the functionals used in research and makes available the Fortran source code that calculates the values of functionals and the necessary derivatives. If one is interested in evaluating other functional derivatives, the book by Parr and Yang [86] includes an appendix on how to perform functional derivatives and the article by Jemmer and Knowles [102] describes a Mathematica program that calculates functional derivatives and gives the Becke88 functional and LYP functional as examples. The program can be obtained online at http://www.tc.bham.ac.uk/~peterk/papers/fderiv/ .
Currently the default functional used by freeMQM-1.cpp is BLYP, which consists of the Becke88 (B88) exchange functional and the LYP correlation functional. Just as an example, the function that contributes to the B88 exchange functional can be written as
The relevant derivatives of
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From the self-consistent-field (SCF) method we obtain the Kohn-Sham (KS) orbitals,
electron density, total ground state energy (
Table 6.1. Density-functional theory results for the helium atom.
The energy values are in atomic units (hartrees).
^{[a] }From Umrigar and Gonze [104]. ^{[b] }Experimental results from the NIST Atomic Spectra Database (ASD). ^{[c] }Computed using freeMQM-1 on a PII 266 MHz with 224 RAM (with ~ 50% of the resources available). ______________________________________________________________________________________________________________________ |
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Our first example of density-functional theory applied to a molecular system is again the hydrogen molecule.
Gill reported [85] a total energy of E = -1.165182 a.u. for H_{2}
using BLYP/6-31G* and the "standard-grid". Note again in Table 3 the large discrepancy between the
the Kohn-Sham orbital energy (e.g.
Table 6.2. Density-functional results for the hydrogen molecule.
The energy values are in atomic units (hartrees).
^{[a] }The experimental value of the bond distance can be found on page 201 of Szabo
and the experimental value of
^{[b] }The exact value of E_{total} can be found in Lowe, p.232. ^{[c] }Computed using freeMQM-1 on a PII 266 MHz with 224 RAM (with ~ 50% of the resources available). __________________________________________________________________________________________________________________________________________ |
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Only LD0194 is currently used in xyz_Integrate, but all the sets of quadrature weights and coordinates described by Lebedev and Laikov in Ref. [105] are available in Lebedev-Laikov.cpp. The function xyz_Integrate could easily be generalized to enable it to access all the functions in Lebedev-Laikov.cpp.
Currently the weight functions w_{A}(r) are only calculated for diatomic molecules.
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