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CHEM 938: Density Functional Theory

CHEM 938: Density Functional Theory. Basics of DFT. January 21, 2010. Density Functional Theory. density functional theory treats the electron density quantum mechanically. currently density functional theory (DFT) calculations are the most common type of quantum chemical calculation.

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CHEM 938: Density Functional Theory

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  1. CHEM 938: Density Functional Theory Basics of DFT January 21, 2010

  2. Density Functional Theory density functional theory treats the electron density quantum mechanically • currently density functional theory (DFT) calculations are the most common type of quantum chemical calculation • generally provides the best balance of accuracy and computational efficiency Canadians have made significant contributions to DFT: Walter Kohn (B.Sc., University of Toronto) • 1998 Nobel prize for development of DFT [Rev. Mod. Phys., 71, 1253 (1999)] • considered ‘father’ of DFT Axel Becke (Dalhousie, formerly Queen’s) • developed density functionals and other methods that dramatically improved the accuracy of DFT for chemical systems Tom Ziegler (University of Calgary) • developed various areas of DFT to make it a practical research tool • one of the first researchers to apply DFT to real-world problems

  3. Density versus Wavefunction Wavefunction: • contains all of the information about the system • not an observable quantity • function of 4N variables (3 spatial, 1 spin per electron) Density: • function of only 3 spatial variables • physically measurable quantity • connected to the wavefunction • related to the wavefunction by: does the electron density contain the same information that is contained in the wavefunction?

  4. Density as the Basic Variable the density can be used to obtain all quantities in the Hamiltonian • we need: 1. number of electrons 2. number of nuclei 3. positions of nuclei 4. charges of nuclei • density exhibits ‘cusps’ at nuclear positions • density at the cusps contains all the information needed to determine ZI the Hamiltonian (and hence density) contains all the information needed to determine the ground and excited states

  5. Density as the Basic Variable the density can be used to obtain all quantities in the Hamiltonian 1. number of electrons 2. number of nuclei 3. positions of nuclei 4. charges of nuclei

  6. 1stHohenberg-Kohn Theorem the ground state energy of a system is uniquely determined by the ground state electron density consider two different Hamiltonians: the two Hamiltonians differ in the electron-nuclear attraction term: • electron-nuclear attraction is called external potential in DFT lingo • each Hamiltonian has a unique ground state wavefunction and ground state energy Hohenberg, Kohn, Phys. Rev., 136, B864 (1964)

  7. 1stHohenberg-Kohn Theorem can two different external potentials give the same ground state electron density?? based on the variational principle: we know: Hohenberg, Kohn, Phys. Rev., 136, B864 (1964)

  8. 1stHohenberg-Kohn Theorem can two different external potentials give the same ground state electron density?? if we do the same for E0,b: in both cases we assumed that: Hohenberg, Kohn, Phys. Rev., 136, B864 (1964)

  9. 1stHohenberg-Kohn Theorem can two different external potentials give the same ground state electron density?? + 0 clearly, the sum of two energies cannot be less than the sum of the same two energies therefore, our assumption that two different wavefunctions with different energies can give the same density is wrong the ground state energy of a system is uniquely determined by the ground state electron density Hohenberg, Kohn, Phys. Rev., 136, B864 (1964)

  10. 1stHohenberg-Kohn Theorem the ground state energy of a system is uniquely determined by the ground state electron density the first Hohenberg-Kohn theorem is an existence theorem • it tells us that a unique relationship exists between the ground state energy and the ground state electron density • it does not tell us what that relationship is we do know that the energy is a functional of the density • a functional is a function whose argument is another function • we will look at the mathematics of functionals in more detailed later • it can be shown that all other ground state properties of the system can be expressed as functionals of the electron density, too Hohenberg, Kohn, Phys. Rev., 136, B864 (1964)

  11. Hohenberg-Kohn Theorems Hohenberg and Kohn proved that the ground state energy is a functional of the density energy functional: • accounts for energy terms only involving electrons HK functional: • kinetic energy of the electrons • electron-electron interactions (Coulomb and non-classical) • if FHK was known exactly, we could get the exact energy of the system Hohenberg, Kohn, Phys. Rev., 136, B864 (1964)

  12. 2ndHohenberg-Kohn Theorem the ground state density is a variational quantity Proof: • we know there is unique correspondence between the density and the wavefunction • so, if we pick an arbitrary trial density, we will get an arbitrary trial wavefunction • the variational theory tells us that for any trial wavefunction: This means: • if we pick any arbitrary density it will give us an upper bound on the true ground state density • if we compare two densities, the one with the lower energy is the better one • we can employ linear variation techniques to optimize the density, like we did with wavefunction methods Hohenberg, Kohn, Phys. Rev., 136, B864 (1964)

  13. Constraints on the Density the wavefunction must satisfy the following constraints: • wavefunctionmust satisfy Pauli exclusion principle • wavefunctionmust be antisymmetric • wavefunctionmust not distinguish between electrons the density must satisfy the following constraints: N-representability: density corresponds to an antisymmetricwavefunction V-representability: density corresponds to an antisymmetricwavefunction of some Hamiltonian, H, with some external potential, υ Levy, PNAS, 76, 6062 (1979)

  14. Constrained Search Method Levy suggested that we search through all densities to find the ground state density - impossible, but useful in theory search works in two steps: 1. search over all wavefunctions that yield as particular density • constrain density to integrate to correct number of electrons, N 2. repeat step 1 for every possible density • constrain densities to be v-representable density that yields the lowest energy is the correct one when the correct density if found, the function F[ρ] = FHK[ρ] Levy, PNAS, 76, 6062 (1979)

  15. Energy Functionals First HK Theorem: • there exists an exact functional relationship between the ground state density and the ground state energy Second HK Theorem: • the electron density can be optimized variationally to use variational optimization procedures, we must know how E and  are related!!! • unfortunately, the form of the exact energy functional remains unknown • regardless, people have developed approximate energy functionals 1. Thomas-Fermi model 2. Kohn-Sham model

  16. Thomas-Fermi Energy Functional in 1927 Thomas and Fermi tried DFT (this was 37 years before Hohenberg and Kohn’s theorems) they assumed the total energy is a sum of kinetic and potential energies they wrote the potential energy as: nuclear-electron Coulomb attraction electron-electron Coulomb repulsion Thomas, Proc. Cambridge Phil. Soc., 23, 542 (1927) Fermi, Rend. Accad. Naz. Lincei, 6, 602 (1927)

  17. Thomas-Fermi Energy Functional in 1927 Thomas and Fermi tried DFT (this was 37 years before Hohenberg and Kohn’s theorems) they assumed the total energy is a sum of kinetic and potential energies they wrote the kinetic energy as: • this is the kinetic energy of an ideal electron gas, a system in which electrons move against a background of uniformly distributed positive charge • the ideal electron gas model doesn’t sound much like a molecule Thomas, Proc. Cambridge Phil. Soc., 23, 542 (1927) Fermi, Rend. Accad. Naz. Lincei, 6, 602 (1927)

  18. Thomas-Fermi Energy Functional in 1927 Thomas and Fermi tried DFT (this was 37 years before Hohenberg and Kohn’s theorems) they assumed the total energy is: we can use this to variationally optimize the density: • and we will find out that the model fails miserably • for example, the TF energy functional predicts that molecules are unstable with respect to the atoms that form them Thomas, Proc. Cambridge Phil. Soc., 23, 542 (1927) Fermi, Rend. Accad. Naz. Lincei, 6, 602 (1927)

  19. Thomas-Fermi Energy Functional what’s wrong with the Thomas-Fermi functional? 1. kinetic energy: • based on an ideal electron gas model that doesn’t even resemble a molecule • but how do you express the kinetic energy in terms of ? 2. electron-electron repulsion: • electron repulsion is treated as the average Coulomb interaction • fails to account for quantum mechanical effects like exchange • fails to account for instantaneous electron-electron correlation Thomas, Proc. Cambridge Phil. Soc., 23, 542 (1927) Fermi, Rend. Accad. Naz. Lincei, 6, 602 (1927)

  20. Kohn-Sham Energy Functional Kohn and Sham suggested decomposing the total energy into kinetic and potential energy contributions • accounts for the kinetic energy of the electrons • accounts for nuclear-electron Coulombic attraction • accounts for average electron-electron Coulombic repulsion • accounts for all electron-electron interactions not in • includes exchange and instantaneous electron-electron correlation Kohn, Sham Phys. Rev., 140, A1133 (1965)

  21. Kohn-Sham Energy Functional Kohn and Sham recognized that it’s easier to calculate the kinetic energy if we have a wavefunction but we don’t know the ground state wavefunction (if we did, we wouldn’t bother with DFT) So, they suggested: • instead of using the real wavefunction and density, let’s use a Slater determinant wavefunction built up from a set of one-electron orbitals (like molecular orbitals) to build up an artificial electron system that represents the ground state density • ‘artificial system’ is often called the non-interacting reference system • the ‘one-electron orbitals’ are called Kohn-Sham orbitals • the Kohn-Sham orbitals are orthonormal: Kohn, Sham Phys. Rev., 140, A1133 (1965)

  22. Kohn-Sham Energy Functional the Kohn-Sham orbitals give us an easy way to construct the density and get an approximate value of the kinetic energy kinetic energy of N one-electron orbitals: density from N one-electron orbitals: the density integrates to give the total number of electrons: Kohn, Sham Phys. Rev., 140, A1133 (1965)

  23. Kohn-Sham Energy Functional a Slater determinant wavefunction built up from Kohn-Sham orbitals will never give the exact ground state kinetic energy (we’ve seen that the closest approximation to the real wavefunction you can get with a single Slater determinant is the Hartree-Fock wavefunction) so, Kohn and Sham suggested splitting up the kinetic energy correction accounting for kinetic energy not contained in Trs exact ground state kinetic energy kinetic energy of the reference system Kohn, Sham Phys. Rev., 140, A1133 (1965)

  24. Kohn-Sham Energy Functional incorporating the orbital-based expressions into the total energy functional: gives: this is the Kohn-Sham total energy functional Kohn, Sham Phys. Rev., 140, A1133 (1965)

  25. Kohn-Sham Energy Functional with Kohn-Sham orbitals, we can calculate: we don’t have an expression for Exc: • Exc is called the exchange correlation functional • it acts as a repository for all contributions to the energy that we do not know how to calculate exactly • if we had an exact expression for Exc we could calculate the energy and density exactly

  26. The Exchange-Correlation Functional the exchange-correlation functional includes all contributions to the energy that are not included in the other terms in the Kohn-Sham energy functional 1. exchange energy: • the classical electron-electron energy term does not account for Pauli repulsion between electrons of the same spin 2. correlation energy: • the classical electron-electron energy term does not account for instantaneous Coulomb interactions between electrons 3. self-interaction energy: • the classical electron-electron energy term includes a spurious contribution from each electron interacting with itself 4. kinetic energy correction: • the kinetic energy of the reference system is not equal to the ground state kinetic energy of the real system Perdew et al., J. Chem. Phys., 123, 062201 (2005)

  27. Self-Interaction Energy For the Hydrogen atom: • there is only one electron • we represent the electron with a probability distribution and average it over all space • so the electron makes a contribution to the density at r1 and r2 • leads to a spurious electron repulsion • the situation extends to multi-electron systems electron density of the hydrogen atom • the self-interaction error must be cancelled out • in Hartree-Fock, exchange cancels out the self-interaction • in DFT, we use the exchange-correlation functional to cancel out the self-interaction

  28. The Exchange-Correlation Functional the 1st Hohengberg-Kohn theorem tells us that an exact functional relationship between energy and the density this means an exact form of Exc must exist, and if we had it, the total energy functional would be exact unfortunately, we do not know the exact form of Exc • the accuracy of DFT calculations hinges on the accuracy various approximations to Exc • the development of exchange-correlation functionals is a major area of research in modern theoretical chemistry • we’ll talk about about exchange-correlation functionals in greater detail later • for now, let’s just treat the exchange-correlation functional in a generic sense

  29. The Kohn-Sham Orbitals how do we solve for the Kohn-Sham orbitals? the second Hohenberg-Kohn theorem tells us that the ground state density minimizes the energy this means: • we need connections between the energy, orbitals, and density • we need a way to ‘optimize’ the orbitals  we’ll use variational theory a variational treatment of the orbitals will involve • expanding the Kohn-Sham orbitals as linear combinations of basis functions: basis function with a fixed form coefficient in linear expansion artificial spin function • using linear variation methods to get the set of coefficients that give us a density that minimizes the energy

  30. The Kohn-Sham Orbitals how do we solve for the Kohn-Sham orbitals? in terms of Kohn-Sham orbitals:

  31. The Kohn-Sham Orbitals how do we solve for the Kohn-Sham orbitals? we have connections between the orbitals, density and energy: we also have a connection to the orbital coefficients:

  32. The Kohn-Sham Orbitals we have to apply the apply the linear variational approach to the Kohn-Sham DFT energy with the constraint that the Kohn-Sham orbitals remain orthonormal if you do this (we won’t), you find out that the ‘best’ set of Kohn-Sham orbitals are given by: energy of Kohn-Sham orbital i Kohn-Sham orbital i Kohn-Sham operator

  33. The Kohn-Sham Operator the Kohn-Sham orbitals are eigenfunctions of the Kohn-Sham operator energy of Kohn-Sham orbital i Kohn-Sham orbital i Kohn-Sham operator the first three terms account for: • the kinetic energy of the electron in i • the Coulombic attraction between the nuclei and the electron in i • the static Coulombic repulsion between the electron in i and the total electron density the last term accounts for: • exchange repulsion, electron correlation, self-interaction energy and kinetic energy correction associated with the electron in i

  34. Comparison with Hartree-Fock the Kohn-Sham orbitals are eigenfunctions of the Kohn-Sham operator the Hartree-Fock molecular orbitals are eigenfunctions of the Fock operator these two expressions are very similar: • the kinetic energy, nuclear-electron, and static Coulomb electron-electron repulsion terms in each operator are identical (but they may be written slightly differently) • the last terms differ: • in the Fock operator, only exchange interactions are considered  can never give exact results • in the Kohn-Sham operator, all remaining interactions are considered  can give exact results, if Vxc is exact

  35. Solving for the Kohn-Sham Orbitals using a linear combination of basis functions we want to solve this is completely analogous to what we did in the Hartree-Fock method multiply on the left by * and integrate: FKS is an element of the Kohn-Sham matrix S is an element of the overlap matrix Note: there will be K of these equations because there are K basis functions

  36. 1 2 K Solving for the Kohn-Sham Orbitals by converting to a matrix form, we can solve for the orbitals with a self-consistent procedure like we did with Hartree-Fock FKSC = SC  is a diagonal matrix of the orbital energies, i: C is a K x K matrix whose columns define the coefficients, ci:

  37. Solving for the Kohn-Sham Orbitals once again, the Kohn-Sham matrix elements depend on the Kohn-Sham orbitals coefficients P is the density matrix, it quantifies the amount of electron density ‘shared’ between basis functions  and 

  38. Solving for the Kohn-Sham Orbitals once again, the Kohn-Sham matrix elements depend on the Kohn-Sham orbitals coefficients Solution: • solve with iterative techniques • make an initial guess of C1 • build FKS1 and solve for C2 • use C2 tobuild FKS2 and solve for C3 . . . • use CN tobuild FKSN and solve for CN+1 • stop when CN and CN+1 are the same (or very similar) • this self-consistent field (SCF) approach was also used in Hartree-Fock • final matrix C defines the Kohn-Sham orbitals that give the density that minimizes the energy

  39. Kohn-Sham Density Functional Theory isn’t this just Hartree-Fock? No, in Hartree-Fock we try to solve the Schrödinger equation using a trial wavefunction consisting of a single Slater determinant since the Hartree-Fock wavefunction is necessarily approximate, this can never give the exact ground state energy

  40. Kohn-Sham Density Functional Theory isn’t this just Hartree-Fock? in Kohn-Sham DFT we only use the Kohn-Sham orbitals to give us the density: that minimizes the Kohn-Sham energy functional: if we have an exact form of Exc, the total energy functional is exact and we can get the exact ground state energy and density with DFT

  41. Kohn-Sham Orbitals do the Kohn-Sham orbitals mean anything? • strictly, the Kohn-Sham orbitals do not have a physical meaning: they are only a way to get at the density • in practice they can be very useful for interpreting the electronic structure • they benefit from being obtained through a one-electron operator that is includes correlation effects • but, you should employ caution when using DFT orbitals to evaluate things like HOMO-LUMO gaps, etc. B3LYP HOMO of ethene Hartree-Fock HOMO of ethene

  42. Performance Overview how does Kohn-Sham DFT do in terms of effort? constructing the Coulomb terms requires the most effort:

  43. Performance Overview how does Kohn-Sham DFT do in terms of effort? constructing the Coulomb terms requires the most effort: there will formally be K4 combinations of basis functions, therefore DFT can scale as bad as K4

  44. Performance Overview how does Kohn-Sham DFT do in terms of effort? but, we project the density onto a set of auxiliary basis functions: ω = auxiliary basis function there will formally be LK2 combinations of basis functions, therefore DFT actually scales as LK2 ~ K3

  45. Performance Overview compare with Hartree-Fock in Hartree-Fock, we have to evaluate the exchange terms: • we cannot apply auxiliary basis functions to the exchange term in Hartree-Fock • instead, in Hartree-Fock we have to calculate all K4 combinations • therefore, Hartree-Fock scales as K4 take home message: DFT scales with system size better than Hartree-Fock • in practice, we say the two methods are similar in cost

  46. Performance Overview what about accuracy? Average errors (kcal/mol) on atomization energies Average errors (Å) on bond lengths method error method error HF/6-31G(d) HF/6-31G(d,p) 85.9 0.021 HF/6-311G(2df,p) HF/6-311G(d,p) 82.0 0.022 MP2/6-31G(d) MP2/6-31G(d,p) 22.4 0.014 MP2/6-31G(d,p) MP2/6-311G(d,p) 23.7 0.014 QCISD/6-31G(d) QCISD/6-311G(d,p) 28.8 0.013 CCSD(T)/6-311G(2df,p) CCSD(T)/6-311G(d,p) 11.5 0.013 SVWN/6-31G(d) 35.7 SVWN/6-311G(d,p) 0.017 Å BLYP/6-31G(d) 5.6 SVWN/6-31G(d,p) 0.016 Å BLYP/6-311G(2df,p) 9.6 PBE/6-311G(d,p) 0.012 Å B3LYP/6-31G(d) 5.2 B3LYP/6-311G(d,p) 0.004 Å B3LYP/6-311+G(3df,2p) 2.2 B3LYP/6-31G(d,p) 0.030 Å

  47. Orbital Free DFT the best efficiency can be achieved if we get rid of the orbitals Kohn-Sham DFT: • construct a Slater determinant wavefunction comprising orbitals • wavefunction depends on 4N variables (3 spatial, 1 spin per electron) • use orbitals to get density and other properties Orbital-Free DFT: • uses the density directly • density only depends on 3 spatial variables • in principle, scales linearly with system size (or maybe NlnN) we are going to focus on Kohn-Sham DFT because orbital-free DFT cannot be applied to many systems in its present form

  48. Orbital Free DFT the best efficiency can be achieved if we get rid of the orbitals • still requires use of exchange-correlation functional • also requires kinetic energy density functionals (KEDFs) Zhou, Ligneres, Carter, Journal of Chemical Physics122, 044103 (2005) • existing KEDFs are best-suited to metals • impressive results to date, but still limited in applicability screw and edge dislocations in Fe (> 4000 atoms) dislocation nucleation in Al3Mg (2x2x1 μm) approx. 1 million DFT calculations Shin et al, Phil. Mag., 89, 3195 (2009) Hayes et al, Phil. Mag., 86, 2343 (2006)

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