Density functional theory (quantum physics)

The density functional theory (DFT) is a method for determining the quantum-mechanical ground state of a many-electron system which on the location-dependent electron density based. Density functional theory is used to calculate basic properties of molecules and solids , such as bond lengths and energies .

The great importance of this theory lies in the fact that it is not necessary to solve the complete Schrödinger equation for the many-electron system, which significantly reduces the amount of computing power and makes calculations of systems with well over ten electrons possible in the first place.

In 1998 the Nobel Prize in Chemistry was awarded to Walter Kohn for the development of density functional theory .

Basics

The basis of the density functional theory is the Hohenberg-Kohn theorem : The basic state of a system of electrons (as a wave function, i.e. dependent on coordinates) is clearly defined by the location-dependent electron density . In density functional theory, the electron density in the ground state is now determined. In principle, all other properties of the basic state can be determined from this. These properties, for example the total energy, are therefore functionals of density. The core problem of density functional theory is therefore to find the corresponding functional. ${\displaystyle N}$ ${\displaystyle 3N}$ ${\displaystyle \rho ({\vec {r}})}$

The total energy can be written as the sum of the kinetic energy , the electron-nucleus interaction and the electron-electron interaction : ${\displaystyle \mathbf {T} [\rho ]}$ ${\displaystyle \mathbf {E} _{ne}[\rho ]}$ ${\displaystyle \mathbf {E} _{ee}[\rho ]}$

${\displaystyle \mathbf {E} [\rho ]=\mathbf {T} [\rho ]+\mathbf {E} _{ne}[\rho ]+\mathbf {E} _{ee}[\rho ].}$

The electron-electron interaction itself is also written as the sum of a part for the electrostatic repulsion between the electrons and a term for the non-classical effects, that is, for the exchange interaction (from English "Exchange Correlation" with " " for " e. " x change " and" "for" c orrelation "): ${\displaystyle \mathbf {E} _{ee}[\rho ]}$ ${\displaystyle \mathbf {J} [\rho ]}$ ${\displaystyle \mathbf {E} _{XC}[\rho ]}$ ${\displaystyle x}$ ${\displaystyle c}$

${\displaystyle \mathbf {E} _{ee}[\rho ]=\mathbf {J} [\rho ]+\mathbf {E} _{XC}[\rho ]}$

The functionals for the electron-nucleus interaction and the electrostatic repulsion of the electron-electron interaction (the so-called Coulomb part) can be derived from the corresponding classical expressions. This is not possible for the kinetic energy and the exchange part of the electron-electron interaction. Therefore, approximations must be sought for these terms. Since the kinetic energy makes up a considerable proportion of the total energy, approximations that are too rough can have drastic effects. Earlier approaches with functionals based solely on density , such as the Thomas-Fermi model , were unsuitable for describing molecules. In fact, the Thomas-Fermi model does not predict stable molecules with chemical bonds . ${\displaystyle \mathbf {E} _{ne}[\rho ]}$ ${\displaystyle \mathbf {J} [\rho ]}$ ${\displaystyle \mathbf {E} _{ee}[\rho ]}$ ${\displaystyle \mathbf {T} [\rho ]}$ ${\displaystyle \mathbf {E} _{XC}[\rho ]}$ ${\displaystyle \rho }$

The Kohn Sham Functions

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One problem with the early, purely density-based DFT approaches, such as the Thomas-Fermi model, was the approximation in the functional of the kinetic energy . By using a determinant approach with orbitals (one-electron wave functions ), analogous to the Hartree-Fock theory , the Kohn-Sham approach (named after Walter Kohn and Lu Jeu Sham ) avoids this problem, as the exact form of the functional of the kinetic energy for a Slater determinant (an overall wavefunction that is composed of individual orbitals) can be calculated exactly. ${\displaystyle \mathbf {T} [\rho ]}$

${\displaystyle \mathbf {T} _{SD}=\sum _{i=1}^{N}\langle {\varphi _{i}}|-{\frac {1}{2}}\nabla ^{2}|\varphi _{i}\rangle {}}$

Orbitals, the so-called Kohn-Sham functions, are thus applied. The electron density is obtained from the sum of the electron densities of the Kohn-Sham functions: ${\displaystyle N}$ ${\displaystyle \varphi _{j}}$

${\displaystyle \rho ({\vec {r}})=\sum _{j=1}^{N}{\left|{\varphi _{j}({\vec {r}})}\right|^{2}}}$

This approach can also and easily calculated. However, the use of a Slater determinant also brings with it a problem: The calculation is now carried out for an interaction-free system that is described by a Slater determinant (one speaks of the "reference system") and not for the real system. Since the energy is a functional of the density and therefore depends exclusively on the density, the exact energy can in principle be obtained from any reference system, provided that its electron density is identical to that of the real system. Thus, the use of a non-interacting Kohn-Sham system is valid as long as its density equals the actual density. Of course, this raises the fundamental question of whether the density of any real system can be reproduced by a single Slater determinant. ${\displaystyle \mathbf {J} [\rho ]}$ ${\displaystyle \mathbf {E} _{ne}[\rho ]}$

However, assuming that the exact density can be reproduced by the Kohn-Sham system, there is still the problem that the exact functional is needed. The difference between the exact kinetic energy and the energy calculated with the determinant wave function together with the difference between the exact electron-electron interaction and the classical Coulomb interaction forms the so-called exchange-correlation functional in the Kohn-Sham theory. ${\displaystyle \mathbf {E} _{xc}[\rho ]}$

${\displaystyle \mathbf {E} _{xc}[\rho ]=(\mathbf {T} [\rho ]-\mathbf {T} _{SD}[\rho ])+(\mathbf {E} _{ee}[\rho ]-\mathbf {J} [\rho ])}$

The entire Kohn-Sham functional can thus be written as:

${\displaystyle \mathbf {E} _{KS}[\rho ]=\mathbf {T} _{SD}[\rho ]+\mathbf {E} _{ne}[\rho ]+\mathbf {J} [\rho ]+\mathbf {E} _{xc}[\rho ]}$

This means that all terms that cannot be precisely determined are found in the exchange-correlation functional and it is therefore not surprising that the individual approaches of modern density functional theory differ primarily in the definition of the exchange-correlation functional. Since the exact form of is not known and it has to be approximated accordingly, the Kohn-Sham approach seems at first glance only to postpone the problem. However, this is deceptive, because while in the original orbital-free approaches the functional of the kinetic energy had to be approximated, which led to serious errors (see above), in the Kohn-Sham approach the kinetic energy for the non-interacting reference system is precisely calculated and only the correction of the kinetic energy and the exchange have to be approximated, which is less serious and gives much better results. ${\displaystyle \mathbf {E} _{xc}[\rho ]}$

The Kohn-Sham orbitals can be obtained as solutions of a system of equations with an effective potential function . This method of calculation (analogous to the Hartree-Fock theory) is much less complex than solving the Schrödinger equation with electrons at the same time, because the solutions of a Schrödinger equation are independent of one another . These one-electron Schrödinger equations are also referred to as Kohn-Sham equations: ${\displaystyle N}$ ${\displaystyle v_{\mathrm {eff} }}$ ${\displaystyle N}$

${\displaystyle \left(-{\frac {1}{2}}\nabla ^{2}+v_{\mathrm {eff} }({\vec {r}})-\epsilon _{j}\right)\varphi _{j}({\vec {r}})=0}$

The effective potential depends on the density:

${\displaystyle v_{\mathrm {eff} }({\vec {r}})=v({\vec {r}})+\int {{\frac {\rho ({\vec {r}}')}{\left|{{\vec {r}}-{\vec {r}}'}\right|}}\mathrm {d} ^{3}r'}+v_{\mathrm {xc} }({\vec {r}})}$

The first term,, is the external potential, which essentially describes the attraction of the electrons by the atomic nucleus, and the second term describes the electrostatic interaction of the electrons with one another (Hartree term). The third term , the so-called exchange - correlation potential ( " " for English " e x change" , " " for " c orrelation ") can be calculated from the exchange-correlation functional: ${\displaystyle v({\vec {r}})}$ ${\displaystyle v_{\mathrm {xc} }({\vec {r}})}$ ${\displaystyle x}$ ${\displaystyle c}$

${\displaystyle v_{xc}={\frac {\delta \mathbf {E} _{xc}}{\delta \rho }}}$

Since the effective potential occurs on the one hand in the Kohn-Sham equations and on the other hand depends on the density and thus on the solutions of these equations, the solutions must be found iteratively . The Kohn-Sham equation is solved again with the newly found potential (or a linear combination of the previous and the new potential), a new potential is determined from this, etc., until a stable ( self-consistent ) solution is found (see also Self- Consistent Field Method ). ${\displaystyle v_{\mathrm {eff} }({\vec {r}})}$ ${\displaystyle \rho ({\vec {r}})}$

Strictly speaking, the Kohn-Sham functions are pure arithmetic variables and have no physical meaning on their own. In practice, however, they can often be used as an approximation for actual electron states, and their energies are used, for example, to calculate the band structure . Their shape and their own energy are also used for qualitative considerations (e.g. as frontier orbitals ). ${\displaystyle \epsilon _{j}}$

The exchange-correlation potential

With the Kohn-Sham formalism, the problem of the many-electron system was actually only shifted to the exchange correlation term , and not yet solved. Strictly speaking, it depends on the electron density at all locations and not just at the point and can only be calculated precisely for very few trivial cases. It turns out, however, that it is often sufficient to find an approximate solution for this term: ${\displaystyle v_{xc}({\vec {r}})}$ ${\displaystyle v_{xc}({\vec {r}})}$ ${\displaystyle {\vec {r}}}$

• Local density (engl. Local density approximation , LDA): With this approximation, it is assumed thata function of the electron density at this locationis. In many cases, this method already provides a sufficiently precise solution, especially when the density is approximately the same everywhere anyway, for example for the conduction electrons in a metal. LDA calculations often lead to an overestimation of the bond strengths; the calculated bond lengths are about one to two percent too short ( overbinding ). ${\displaystyle v_{xc}({\vec {r}})}$ ${\displaystyle \rho ({\vec {r}})}$

• generalized gradient approximation (. engl generalized gradient approximation , GGA): There are not only the densitybut also their derivatives with respect to the location ( gradient considered). There are several different methods for this, most of which are named after the authors of the method, for example PW91 for the method presented by Perdew and Wang in 1991. The PBE functional, which is named after the physicists Perdew, Burke and Ernzerhof, should have the same shape as the PW91, but manage with considerably fewer parameters. The PBE is still one of the most commonly used functional units today, although there are already further developments such as the revPBE or the RPBE. ${\displaystyle \rho ({\vec {r}})}$

• Hybrid methods : Here only part of the exchange correlation potential is calculated according to the density functional theory (e.g. with GGA), part is calculated as the exchange energy of the Kohn-Sham functions as in the Hartree-Fock method . These methods are more precise than pure GGA calculations, especially for molecules, but the effort involved in the calculations is significantly higher than for GGA. The most common hybrid process is known as B3LYP.

Most of the limitations and problems in using density functional theory are related to the exchange correlation potential. For example, the different GGA potentials provide binding energies of simple molecules that can differ from each other and from the experimental values ​​by more than 20 percent. Van der Waals bonds are not described at all by the “semi-local” functions such as GGA, because they are based on long-range correlations of the charge distribution. Another problem is that the band gaps and HOMO - LUMO energy differences, which are calculated from the Kohn-Sham functions, are generally too low for LDA and GGA.

The calculations with the density functional theory are usually carried out in the Born-Oppenheimer approximation , so only the electrons are treated quantum mechanically.

Calculation method on the computer

Calculating the properties of complex molecules or crystals using density functional theory is time-consuming, so the efficiency of the calculations plays a major role. The calculation methods can be classified according to the basic functions for the Kohn-Sham equations:

Atomic wave functions ( muffin-tin orbitals ) in a spherical environment around the atomic nucleus (so-called muffin-tin area) are well suited for describing the electrons near the nucleus. The advantage of atomic wave functions is that for problems that are suitable for their application, very small basic sets (one function per electron and angular momentum character) are usually sufficient for the description. However, problems arise in consistently describing the almost free electrons between the atoms (e.g. conduction electrons in metals, electrons on surfaces, etc.) or the overlap area between the atoms.

Plane waves are well suited for describing the valence electrons and conduction electrons in solids, but the spatially less extensive wave functions close to the atomic nucleus are difficult to describe. Plane waves have the advantage that efficient algorithmscan be usedfor the Fourier transformation and thus the solution of the Kohn-Sham equations can be carried out very quickly. In addition, they are very flexible because, for example, almost free electrons on surfaces can also be described well.

Therefore, especially for calculations in solid-state physics, these methods are combined by using plane waves, but taking additional measures for the area near the atomic nuclei. This area can either be treated completely separately ( augmented plane waves ), additional wave functions can be added there ( projector augmented waves ) or a so-called pseudopotential can be used , which only results in correct wave functions in the area of ​​the outer electrons, but not in the vicinity the atomic nuclei.

For the treatment of molecular systems, atom-centered Gaussian functions are usually used as the basis for the Kohn-Sham orbitals to be generated. These functions are pre-optimized for each atom and are based on the analytically known solutions of the electronic wave functions of the hydrogen atom. These atomic orbitals (one-electron wave functions) are used for the construction of molecular orbitals , taking their symmetry into account by means of suitable linear combinations ( LCAO approach ). By self-consistent solving of the KS equations (effective one-electron Schrödinger equations) one obtains a set of KS orbitals as eigenfunctions of the KS operators, which are summarized in the form of an ( antisymmetrical ) Slater determinant . This Slater determinant only serves to construct the correct electron density and does not represent a reasonable wave function. From this, the kinetic energy of the electrons, the external potential and the classic Coulomb interaction of the electrons can be calculated. Up to this point, the DFT method does not differ significantly from a wave function method. The modern KS-DFT therefore benefits from the long researched and efficiently implemented Hartree-Fock machinery. The significant difference in DFT lies in the calculation of the exchange-correlation energy. This is usually calculated numerically for selected grid points, since modern density functionals sometimes take on very absurd forms and an analytical treatment is therefore very difficult.

With powerful computers, systems of up to around 1000 atoms can now be treated using DFT calculations. For larger systems, other approximation methods such as the tight binding method or approximation methods based on DFT results must be used.

Applications in chemistry

DFT methods can - just like traditional quantum chemical ab initio methods - be used to determine the structure of polyatomic systems. If the atomic positions, nuclear charges and the number of electrons are known, the total energy of the system can be calculated for a given nuclear geometry with the help of the Kohn-Sham formalism. In this way, potential energy surfaces can be mapped and minimum or transition state geometries can be characterized using a so-called geometry optimization . Since the density functional theory only represents an approximation for solving the exact Schrödinger equation in practice , errors inevitably arise. In general, however, these are on the one hand largely systematic (depending on the type of bond and the selected functional) and on the other hand well documented and quantifiable. Even if the systematic ability to improve accuracy is less pronounced than with ab initio methods, a kind of hierarchy of DFT methods has been established with the so-called Jacob's Ladder .

Furthermore, a number of spectroscopic properties of molecules can be calculated or even entire spectra can be predicted. Careful experimental data are used, among other things, to calibrate DFT methods. The density functional theory can therefore be used for the verification or interpretation of existing data or even for predictions. However, the latter often requires the use of empirical correction schemes in order to achieve reliable results. Some exemplary (spectroscopic) applications of DFT can be found in the following list:

• Since the principle of density functional theory can also be extended to excited electronic states (time-dependent density functional theory, TDDFT), potential energy surfaces of electronically excited states and thus electronic excitation energies for optical spectroscopy can also be calculated.
• The electrical dipole moment , which is important for rotational spectroscopy, can be determined directly from the electron density for a fixed core geometry .
• The second derivative of the energy according to an internal coordinate gives the force constant with regard to the deflection. From this, oscillation frequencies can be calculated.
• Magnetic properties can also be represented with modern programs based on DFT - thus EPR and NMR spectra are also accessible.
• Ionization energies and electron affinities can also be calculated via the electronic structure , which are required, for example, to predict photoelectron spectra .

Using statistical models based on the results of a DFT calculation, thermodynamic quantities such as the enthalpy or entropy of a system are also accessible. In addition, thermochemical quantities such as atomization, reaction and binding energies can be calculated. DFT methods are also used in the determination of molecular structures in solids, which can be used for computer-aided crystal structure analysis.

In solid-state physics and solid-state chemistry , density functional theory enables diffusion mechanisms to be calculated. In the nuclear solid state spectroscopy can with its electric field gradients and magnetism can be calculated directly with measurements from Mössbauer spectroscopy , Impaired gamma-gamma angular correlation , nuclear magnetic resonance spectroscopy or Myonenspinspektroskopie can be compared in order to develop models for the local structure in functional materials.

Extensions

There are numerous extensions to the theory, e.g. B. spin density or current density functional theories, or so-called dynamic density functional theories, all of which are worth mentioning, but cannot be discussed here in detail, especially since the area is still very fluid.

literature

• RM Dreizler, Eberhard KU Gross : Density Functional Theory. Springer, Berlin 1990, ISBN 3-540-51993-9 .
• Klaus Capelle: A bird's-eye view of density-functional theory . In: Brazilian Journal of Physics . tape 36 , 4a, December 2006, doi : 10.1590 / S0103-97332006000700035 , arxiv : cond-mat / 0211443v5 .
• Wolfram Koch, Max C. Holhausen: A Chemist's Guide to Density Functional Theory , WILEY-VCH Verlag GmbH, Weinheim, 2001

Weblinks (English)

• Walter Kohn: Nobel Prize Lecture. (PDF; 2.24 MB)
• Hands-on tutorial course on VASP. February 10 to February 13, 2003. Retrieved September 22, 2010 (Another collection of lecture materials from a seminar on the DFT program package VASP).
• FLEUR - Full Potential Linarized Augmented Plane Wave (FLAPW) Code. Research center Julich. Retrieved September 22, 2010.
• O. Jepsen, OK Andersen: Stuttgart TB-LMTO Program. Max Planck Institute for Solid State Research. Retrieved September 22, 2010 (Linear Muffin Tin Orbitals (LMTO) Code).

Individual evidence

1. ↑ ^{a b c d e} A Chemist's Guide to Density Functional Theory, 2nd Edition. October 15, 2001, Retrieved November 30, 2018 (American English). 
2. ↑ Kieron Burke: Perspective on density functional theory . In: The Journal of Chemical Physics . tape 136 , no. 15 , April 17, 2012, ISSN  0021-9606 , p. 150901 , doi : 10.1063 / 1.4704546 ( scitation.org [accessed November 30, 2018]).
3. ^ The Nobel Prize in Chemistry 1998. Retrieved November 30, 2018 (American English). 
4. ↑ P. Hohenberg, W. Kohn: Inhomogeneous Electron Gas . In: Physical Review . tape 136 , 3B, November 9, 1964, pp. B864 – B871 , doi : 10.1103 / PhysRev.136.B864 ( aps.org [accessed November 30, 2018]).
5. ^ Frank Jensen: Introduction to Computational Chemistry, Third Edition . ISBN 978-1-118-82599-0 , pp. 273 .
6. ↑ Axel D. Becke: Perspective: Fifty years of density-functional theory in chemical physics . In: The Journal of Chemical Physics . tape 140 , no. 18 , May 14, 2014, ISSN  0021-9606 , p. 18A301 , doi : 10.1063 / 1.4869598 .
7. ^ ^{A b} W. Kohn, LJ Sham: Self-Consistent Equations Including Exchange and Correlation Effects . In: Physical Review . tape 140 , 4A, November 15, 1965, pp. A1133 – A1138 , doi : 10.1103 / PhysRev.140.A1133 ( aps.org [accessed November 30, 2018]).
8. ↑ John P. Perdew, JA Chevary, SH Vosko, Koblar A. Jackson, Mark R. Pederson: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation . In: Physical Review B . tape 46 , 1992, pp. 6671 , doi : 10.1103 / PhysRevB.46.6671 ( harvard.edu [accessed September 28, 2019]).
9. ↑ John P. Perdew, Kieron Burke, Yue Wang: Generalized gradient approximation for the exchange-correlation hole of a many-electron system . In: Physical Review B . tape 54 , no. 23 , December 15, 1996, pp. 16533-16539 , doi : 10.1103 / PhysRevB.54.16533 .
10. ↑ John P. Perdew, Kieron Burke, Matthias Ernzerhof: Generalized Gradient Approximation Made Simple . In: Physical Review Letters . tape 77 , no. 18 , October 28, 1996, pp. 3865–3868 , doi : 10.1103 / PhysRevLett.77.3865 ( aps.org [accessed September 28, 2019]).
11. ↑ Chengteh Lee, Weitao Yang, Robert G. Parr: Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density . In: Physical Review B . tape 37 , no. 2 , January 15, 1988, p. 785–789 , doi : 10.1103 / PhysRevB.37.785 ( aps.org [accessed September 28, 2019]).
12. ^ AD Becke: Density-functional exchange-energy approximation with correct asymptotic behavior . In: Physical Review A . tape 38 , no. 6 , September 1, 1988, pp. 3098–3100 , doi : 10.1103 / PhysRevA.38.3098 ( aps.org [accessed September 28, 2019]).
13. ↑ Axel D. Becke: Density ‐ functional thermochemistry. III. The role of exact exchange . In: The Journal of Chemical Physics . tape 98 , no. 7 , April 1, 1993, ISSN  0021-9606 , pp. 5648-5652 , doi : 10.1063 / 1.464913 ( scitation.org [accessed September 28, 2019]).
14. ↑ John P. Perdew, Karla Schmidt: Jacob's ladder of density functional approximations for the exchange-correlation energy . In: AIP Conference Proceedings . AIP, 2001, doi : 10.1063 / 1.1390175 ( scitation.org [accessed November 30, 2018]).
15. ↑ Holthausen, Max C .: A chemist's guide to density functional theory . 2nd ed. Wiley-VCH, Weinheim 2001, ISBN 978-3-527-30372-4 .