密度泛函理論

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2个分类: Quantum mechanics | Quantum chemistry

这个条目的部分内容来自量子化学维基（http://wiki.quantumchemistry.net）遵循GFDL协议释出，关于本条目的原文链接请参见这里。

密度泛函理论, Density functional theory (DFT) 是一种研究多电子体系电子结构的量子力学方法。密度泛函理论在物理和化学上都有广泛的应用，特别是用来研究分子和凝聚态的性质，是凝聚态物理和计算化学领域最常用的方法之一。

[编辑] 理论概述

电子结构理论的经典方法，特别是Hartree-Fock方法和后Hartree-Fock方法，是基于复杂的多电子波函数的。密度泛函理论的主要目标就是用电子密度取代波函数做为研究的基本量。因为多电子波函数有 $3 N$ 个变量（ $N$ 为电子数，每个电子包含三个空间变量），而电子密度仅是三个变量的函数，无论在概念上还是实际上都更方便处理。

虽然密度泛函理论的概念起源于Thomas-Fermi模型，但直到Hohenberg-Kohn定理提出之后才有了坚实的理论依据。Hohenberg-Kohn第一定理指出体系的基态能量仅仅是电子密度的泛函。

Hohenberg-Kohn第二定理证明了以基态密度为变量，将体系能量最小化之后就得到了基态能量。

最初的HK理论只适用于没有磁场存在的基态，虽然现在已经被推广了。最初的Hohenberg-Kohn定理仅仅指出了一一对应关系的存在，但是没有提供任何这种精确的对应关系。正是在这些精确的对应关系中存在着近似（这个理论可以被推广到时间相关领域，从而用来计算激发态的性质[6]）。

密度泛函理论最普遍的应用是通过Kohn-Sham方法实现的。在Kohn-Sham DFT的框架中，最难处理的多体问题（由于处在一个外部静电势中的电子相互作用而产生的）被简化成了一个没有相互作用的电子在有效势场中运动的问题。这个有效势场包括了外部势场以及电子间库仑相互作用的影响，例如，交换和相关作用。处理交换相关作用是KS DFT中的难点。目前并没有精确求解交换相关能 $E X C$ 的方法。最简单的近似求解方法为局域密度近似(LDA)。LDA近似使用均匀电子气来计算体系的交换能（均匀电子气的交换能是可以精确求解的），而相关能部分则采用对自由电子气进行拟合的方法来处理。

自1970年以来，密度泛函理论在固体物理学的计算中得到广泛的应用。在多数情况下，与其他解决量子力学多体问题的方法相比，采用局域密度近似的密度泛函理论给出了非常令人满意的结果，同时固态计算相比实验的费用要少。尽管如此，人们普遍认为量子化学计算不能给出足够精确的结果，直到二十世纪九十年代，理论中所采用的近似被重新提炼成更好的交换相关作用模型。密度泛函理论是目前多种领域中电子结构计算的领先方法。尽管密度泛函理论得到了改进，但是用它来恰当的描述分子间相互作用，特别是范德瓦尔斯力，或者计算半导体的能隙还是有一定困难的。 Its poor treatment of dispersion renders DFT unsuitable (at least when used alone) for the treatment of systems which are dominated by dispersion (e.g. interacting noble gas atoms) or where dispersion competes significantly with other effects (e.g. in biomolecules). The development of new DFT methods designed to overcome this problem, by alterations to the functional or by the inclusion of additive terms, is a current research topic.

[编辑] 早期模型: Thomas-Fermi 模型

密度泛函理论可以上溯到由Thomas和Fermi 在1920年代发展的Thomas-Fermi模型。他们将一个原子的动能表示成电子密度的泛函，并加上原子核-电子和电子-电子相互作用（两种作用都可以通过电子密度来表达）的经典表达来计算原子的能量。

Thomas-Fermi模型是很重要的第一步，但是由于没有考虑Hartree-Fock理论指出的原子交换能，Thomas-Fermi方程的精度受到限制。1928年Dirac在该模型基础上增加了一个交换能泛函项。

然而，在大多数应用中Thomas-Fermi-Dirac理论表现得非常不够准确。其中最大的误差来自动能的表示，然后是交换能中的误差，以及对电子相关作用的完全忽略。

[编辑] Derivation and formalism

As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (Born-Oppenheimer approximation), generating a static external potential $\,\!V$ in which the electrons are moving. A stationary electronic state is then described by a wave function $\Psi(\vec r_1,\dots,\vec r_N)$ fulfilling the many-electron Schrödinger equation

$H \Psi = \left[{T}+{V}+{U}\right]\Psi = \left[\sum_i^N -\frac{\hbar^2}{2m}\nabla_i^2 + \sum_i^N V(\vec r_i) + \sum_{i<j}U(\vec r_i, \vec r_j)\right] \Psi = E \Psi$

where $\,\!N$ is the number of electrons and $\,\!U$ is the electron-electron interaction. The operators $\,\!T$ and $\,\!U$ are so-called universal operators as they are the same for any system, while $\,\!V$ is system dependent or non-universal. As one can see the actual difference between a single-particle problem and the much more complicated many-particle problem just arises from the interaction term $\,\!U$ . Now, there are many sophisticated methods for solving the many-body Schrödinger equation, e.g. there is diagrammatic perturbation theory in physics, while in quantum chemistry one often uses configuration interaction (CI) methods, based on the systematic expansion of the wave function in Slater determinants. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger complex systems.

Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the many-body problem, with $\,\!U$ , onto a single-body problem without $\,\!U$ . In DFT the key variable is the particle density $n(\vec r)$ which is given by

$n(\vec r) = N \int{\rm d}^3r_2 \int{\rm d}^3r_3 \cdots \int{\rm d}^3r_N \Psi^*(\vec r,\vec r_2,\dots,\vec r_N) \Psi(\vec r,\vec r_2,\dots,\vec r_N).$

Hohenberg and Kohn proved in 1964 [1] that the relation expressed above can be reversed, i.e. to a given ground state density $n_0(\vec r)$ it is in principle possible to calculate the corresponding ground state wave function $\Psi_0(\vec r_1,\dots,\vec r_N)$ . In other words, $\,\!\Psi_0$ is a unique functional of $\,\!n_0$ , i.e.

$\,\!\Psi_0 = \Psi_0[n_0]$

and consequently all other ground state observables $\,\!O$ are also functionals of $\,\!n_0$

$\left\langle O \right\rangle[n_0] = \left\langle \Psi_0[n_0] \left| O \right| \Psi_0[n_0] \right\rangle.$

From this follows in particular, that also the ground state energy is a functional of $\,\!n_0$

$E_0 = E[n_0] = \left\langle \Psi_0[n_0] \left| T+V+U \right| \Psi_0[n_0] \right\rangle$ ,

where the contribution of the external potential $\left\langle \Psi_0[n_0] \left|V\right| \Psi_0[n_0] \right\rangle$ can be written explicitly in terms of the density

$V[n] = \int V(\vec r) n(\vec r){\rm d}^3r.$

The functionals $\,\!T[n]$ and $\,\!U[n]$ are called universal functionals while $\,\!V[n]$ is obviously non-universal, as it depends on the system under study. Having specified a system, i.e. $\,\!V$ is known, one then has to minimise the functional

$E[n] = T[n]+ U[n] + \int V(\vec r) n(\vec r){\rm d}^3r$

with respect to $n(\vec r)$ , assuming one has got reliable expressions for $\,\!T[n]$ and $\,\!U[n]$ . A successful minimisation of the energy functional will yield the ground state density $\,\!n_0$ and thus all other ground state observables.

The variational problem of minimising the energy functional $\,\!E[n]$ can be solved by applying the Lagrangian method of undetermined multipliers, which was done by Kohn and Sham in 1965 [2]. Hereby, one uses the fact that the functional in the equation above can be written as a fictitious density functional of a non-interacting system

$E_s[n] = \left\langle \Psi_s[n] \left| T_s+V_s \right| \Psi_s[n] \right\rangle,$

where $\,\!T_s$ denotes the non-interacting kinetic energy and $\,\!V_s$ is an external effective potential in which the particles are moving. Obviously, $n_s(\vec r)\equiv n(\vec r)$ if $\,\!V_s$ is chosen to be

$V_s = V + U + \left(T_s - T\right).$

Thus, one can solve the so-called Kohn-Sham equations of this auxiliary non-interacting system

$\left[-\frac{\hbar^2}{2m}\nabla^2+V_s(\vec r)\right] \phi_i(\vec r) = \epsilon_i \phi(\vec r),$

which yields the orbitals $\,\!\phi_i$ that reproduce the density $n(\vec r)$ of the original many-body system

$n(\vec r )\equiv n_s(\vec r)= \sum_i^N \left|\phi_i(\vec r)\right|^2.$

The effective single-particle potential $\,\!V_s$ can be written in more detail as

$V_s = V + \int \frac{e^2n_s(\vec r\,')}{|\vec r-\vec r\,'|} {\rm d}^3r' + V_{\rm XC}[n_s(\vec r)],$

where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term $\,\!V_{\rm XC}$ is called exchange correlation potential. Here, $\,\!V_{\rm XC}$ includes all the many particle interactions. Since the Hartree term and $\,\!V_{\rm XC}$ depend on $n(\vec r )$ , which depends on the $\,\!\phi_i$ , which in turn depend on $\,\!V_s$ , the problem of solving the Kohn-Sham equation has to be done in a self-consistent way. Usually one starts with an initial guess for $n(\vec r)$ , then one calculates the corresponding $\,\!V_s$ and solves the Kohn-Sham equations for the $\,\!\phi_i$ . From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached.

[编辑] Approximations

The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas. However, approximations exist which permit the calculation of certain physical quantities quite accurately. In physics the most widely used approximation is the local density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:

$E_{XC}[n]=\int\epsilon_{XC}(n){\rm d}^3r.$

The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron spin:

$E_{XC}[n_\uparrow,n_\downarrow]=\int\epsilon_{XC}(n_\uparrow,n_\downarrow){\rm d}^3r.$

Highly accurate formulae for the exchange-correlation energy density $\epsilon_{XC}(n_\uparrow,n_\downarrow)$ have been constructed from simulations of a free-electron gas.

Generalized gradient approximations (GGA) are still local but also take into account the gradient of the density at the same coordinate:

$E_{XC}[n_\uparrow,n_\downarrow]=\int\epsilon_{XC}(n_\uparrow,n_\downarrow,\vec{\nabla}n_\uparrow,\vec{\nabla}n_\downarrow){\rm d}^3r.$

Using the latter (GGA) very good results for molecular geometries and ground state energies have been achieved. Many further incremental improvements have been made to DFT by developing better representations of the functionals.

[编辑] Relativistic generalization

The relativistic generalization of the DFT formalism leads to a current density functional theory.

[编辑] Applications

In practice, Kohn-Sham theory can be applied in two distinct ways depending on what is being investigated. In the solid state, plane wave basis sets are used with periodic boundary conditions. Moreover, great emphasis is placed upon remaining consistent with the idealised model of a 'uniform electron gas', which exhibits similar behaviour to an infinite solid. In the gas and liquid phases, this emphasis is relaxed somewhat, as the uniform electron gas is a poor model for the behaviour of discrete atoms and molecules. Because of the relaxed constraints, a huge variety of exchange-correlation functionals have been developed for chemical applications. The most famous and popular of these is known as B3LYP [3-5]. The adjustable parameters of these functionals are generally fitted to a 'training set' of molecules. Unfortunately, although the results obtained with these functionals are usually relatively accurate for most applications, there is no systematic way of improving them (in contrast to some of the traditional wavefunction-based methods like configuration interaction or coupled cluster method). Hence, in the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiment.

[编辑] References

[1] P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864
[2] W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133
[3] A. D. Becke, J. Chem. Phys. 98 (1993) 5648
[4] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37 (1988) 785
[5] P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, J. Phys. Chem. 98 (1994) 11623

[编辑] Literature

Klaus Capelle, A bird's-eye view of density-functional theory

来自“http://www.wikilib.com/wiki/%E5%AF%86%E5%BA%A6%E6%B3%9B%E5%87%BD%E7%90%86%E8%AB%96”

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