CHEM 938: Density Functional Theory

1. CHEM 938: Density Functional Theory Functionals January 26, 2010

2. Basic Details of DFT recall from last lecture • we want to minimize the Kohn-Sham energy functional: • by optimizing a set of orbitals: • to get at the correct density for the system

3. Basic Details of DFT recall from last lecture • the Kohn-Sham energy function is exact in principle: • because Exc contains all the contributions to the energy not accounted for by the other terms: • electron correlation • exchange energies • self-interaction energies • kinetic energy correction • in practice, we have to use approximations to the exact form of Exc to understand how the total energy is minimized and how Exc is constructed, it is useful to look at exchange and correlation

4. Exchange and Correlation we have already encountered exchange and correlation energies exchange: • correlation of electron motions due to Pauli exclusion principle • only affects electrons of the same spin • captured in the Hartree-Fock method • expressed as: correlation: • all the energy not captured by Hartree-Fock • static correlation accounts for the ability of electrons to avoid each other by moving into degenerate states • dynamic correlation accounts for the ability of electrons to avoid each other by adjusting their motions to minimize Coulomb repulsion • we will call this Coulomb correlation exchange and correlation are two electron interactions beyond the average Coulomb repulsion

5. Exchange and Correlation exchange and correlation are two electron interactions beyond the average Coulomb repulsion • the density doesn’t directly tell us a lot about instantaneous electron-electron interactions • essentially, the density tells us the probability of finding an electron at some point in space, while all the other electrons have arbitrary positions and spins • for two-electron interactions, it is more useful to know the pair density: the pair density tells us the probability of finding electrons at coordinates x1 and x2 while all the other electrons have arbitrary coordinates

6. Pair Density and Reduced Density Matrices pair density contains all the information needed to determine the exchange and correlation energies • it is useful to generalize ρ2 to yield the reduced density matrix: • based on the antisymmetry of the wavefunction: the reduced density matrix is also antisymmetric with respect to particle exchange

7. Pair Density and Reduced Density Matrices pair density contains all the information needed to determine the exchange and correlation energies • the diagonal elements of the reduced density matrix give back the pair density • if we consider the case where x1 = x2: probability of finding two electrons of same spin at same point in space is 0 = Pauli exclusion principle

8. Pair Density and Reduced Density Matrices consider a Hartree-Fockwavefunction for a two-electron system if the electrons have opposite spins: • if the pair density is just a product of individual densities the electron motions are uncorrelated • thus, the pair density correctly tells us that the Hartree-Fockwavefunction does not correlate the motions of electrons with opposite spins • in fact, ρ2(x1,x1) ≠ 0 → electrons of opposite spin can be at same point in space!!

9. Pair Density and Reduced Density Matrices consider a Hartree-Fockwavefunction for a two-electron system if the electrons have the same spins: • the pair density correctly tells us that the Hartree-Fockwavefunction does correlate the motions of electrons with identical spins through exchange • ρ2(x1,x1) = 0 → electrons of identical spin cannot be at same point in space • this is a purely quantum mechanical correlation effect called exchange

10. Pair Density and Reduced Density Matrices consider a Hartree-Fockwavefunction for a two-electron system opposite spins: identical spins: in general: correlation factor • the correlation factor accounts for interactions between electrons that are not captured through the average Coulomb repulsion model • if f(x1;x2) = 0, the average Coulomb repulsion model is recovered • all other forms of the correlation factor introduce some correlation • getting the exact form of f(x1;x2) is the goal in DFT because it would allow us to describe the correlation and exchange energies in terms of the density

11. Pair Density and Reduced Density Matrices let’s look at the pair density in more detail consider the case with f(x1;x2) = 0: • this is the completely uncorrelated case • but, integration yields the wrong number of electron pairs: • extra pairs correspond to double counting of interactions plus self-interaction • so, we define the conditional probability: probability of finding an electron at x2 if one is known to exist at x1

12. Exchange-Correlation Holes we can interpret all non-classical electron-electron interactions in terms of an exchange-correlation hole function hole difference between conditional probability and uncorrelated probability

13. Exchange-Correlation Holes we can interpret all non-classical electron-electron interactions in terms of an exchange-correlation hole function • the hole function accounts for the difference between the completely uncorrelated case and the completely correlated case • these effects are: • self-interaction • exchange (Pauli repulsion) • Coulomb correlation (instantaneous electron-electron interactions) if we can develop an exact expression for the exchange-correlation hole, we can describe electron-electron interactions exactly

14. Exchange-Correlation Holes let’s investigate the exchange-correlation hole in more detail hXC contains 1 electron: • this can be thought of as the electron at x1 ‘digging a hole’ around itself so that the probability of finding another electron nearby is diminished • the amount of electron density ‘dug up’ by the electron is equal to 1 electron

15. Exchange-Correlation Holes let’s investigate the exchange-correlation hole in more detail hXC affects the electron-electron interaction energy:

16. Exchange-Correlation Holes let’s investigate the exchange-correlation hole in more detail hXC affects the electron-electron interaction energy:

17. Exchange-Correlation Holes let’s investigate the exchange-correlation hole in more detail we can split hXC into exchange and correlation parts: • only the total hole is meaningful, but splitting it up helps us construct approximate representations for the hole function • hX is called the Fermi hole and accounts for Pauli repulsion between electrons (this also corrects for the self-interaction) • hC is called the Coulomb hole and accounts for instantaneous electron-electron interactions

18. Fermi Hole Fermi hole accounts for Pauli repulsion • only affects electrons of the same spin • integrates to -1, and is negative in all regions of space: • essentially, reduces the number of electrons the electron at r1 interacts with by 1 • accounts for self-interaction • due to Pauli-repulsion: • depends on density at r2: • not spherically symmetric • usually large at r1 (but doesn’t have to be) • can be ‘left behind’ when ρ(r1) is small • can be very delocalized over a molecule

19. Coulomb Hole Coulomb hole accounts for instantaneous electrostatic repulsion • does not depend on spin • integrates to 0: • essentially, reduces the electron density near the electron at r1 by moving them to other regions of space • the reduction of electron density near r1 cancels the build up of electron density elsewhere, so the change in the number of electron is zero • no clear value as r1 → r2, but must recover ‘cusp’ in density when r1 = r2 (of course electrons with identical spins are also affected by the Fermi hole as r1 → r2)

20. Example consider an H2 molecule Fermi hole Coulomb hole total hole 0.7 Å 1.1 Å 2.6 Å RH-H total hole gives correct behaviour: neither the Fermi nor Coulomb holes alone are sufficient

21. Toward XC Functionals let’s revisit what we actually want to do • we want to get orbitals and a density so we can calculate the energy: • the orbitals are eigenfunctions of the Kohn-Sham operator: • the density is:

22. Toward XC Functionals let’s revisit what we actually want to do • we want to get orbitals and a density so we can calculate the energy: we have known explicit expressions for these terms we can write the functional as an integral containing the functional derivative our goal is to come up with an exact expression (or a good approximate expression) for Vxc

23. Toward XC Functionals what does Vxc represent? • Vxc should contain all the operators needed to get: 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

24. Toward XC Functionals what does Vxc represent? • the Fermi and Coulomb holes clearly account for exchange, self-interaction and electron correlation uncorrelated two-electron energy all non-classical interactions including exchange, correlation and self-interaction correction • what about the correction to the kinetic energy? the hole function also accounts for the kinetic energy error through the adiabatic connection

25. Adiabatic Connection consider the non-interacting and real systems as limits • we can write the Hamiltonian for the system as: • where λ takes on values between 0 and 1 λ-dependent external potential to give density of real system • when λ = 0: • when λ = 1: • varying λ connects these two systems • as λ is varied, Vext is adjusted so that the density always equals the density of the real system

26. Adiabatic Connection consider the non-interacting and real systems as limits • we can think of connecting the energies of the two systems through the following integral: • where dEλis given by:

27. Adiabatic Connection consider the non-interacting and real systems as limits • so: • and: • and, finally: • which we call the coupling-strength integrated exchange-correlation hole now we can calculate the energy of the real system

28. Adiabatic Connection consider the non-interacting and real systems as limits

29. Adiabatic Connection consider the non-interacting and real systems as limits compare with known total energy:

30. Adiabatic Connection consider the non-interacting and real systems as limits so, including the integration over λ accounts for the difference between the kinetic energies of the real and reference systems: • correlation energy • exchange and self-interaction correction • kinetic energy correction if we know the XC hole, we can capture all effects that are supposed to go into the XC functional

31. Toward XC Functionals what does Vxc represent? • Vxc should contain all the operators needed to get: 1. exchange energy: • contained in hx 2. correlation energy: • contained in hC 3. self-interaction energy: • contained in hX 4. kinetic energy correction: • accounted for through adiabatic connection of course, we still need to know how to actually write hXC and VXC

32. Functionals DFT is based on the use of functionals Function: • rule for turning a number into another number f(x) = x2 : x=2 → f(x) = 4 Functional: • rule for turning a function into a number • alternatively, a function whose argument is another function

33. Functionals DFT is based on the use of functionals • the integration is needed because the functional depends on the function at all points in space integrate over spatial coordinates • often, we want to know how the functional changes is we change the function • this can be used to figure out which wavefunction minimizes the energy to do this, we need a functional derivative

34. Variational Energy Minimization • if we restrict Ψ to be a Slater determinant formed from MOs that are linear combinations of basis functions: energy • we change Ψ by changing {cνi}, so we are looking for dE/dcvi = 0 • essentially E = E[Ψ(r)] and Ψ(r) is altered by changing {cνi} {cνi} recall, linear variation: trial wavefunction Hamiltonian and overlap matrix elements energy of trial wavefunction

35. Variational Energy Minimization • if we restrict Ψ to be a Slater determinant formed from MOs that are linear combinations of basis functions: energy • we change Ψ by changing {cνi}, so we are looking for dE/dcvi = 0 • essentially E = E[Ψ(r)] and Ψ(r) is altered by changing {cνi} {cνi} recall, linear variation: ‘best’ trial wavefunction minimizes energy

36. Variational Energy Minimization • if we consider all possible Ψs, we will get many different wavefunctions that are local energy minima energy • global minimum is the real wavefunction • at the minima, dE/dΨ = 0 → this is a functional derivative Ψ functional derivatives also appear in the Kohn-Sham operator:

37. Functional Derivatives how do functionals change when the function changes? • consider a function, f(x): first derivative (slope) of f(x) • consider a functional, F[f(x)]:

38. Functional Derivatives how do functionals we calculate functional derivatives? • we expand out the Taylor series for the functional • consider the Thomas-Fermi kinetic energy functional • the functional derivative is:

39. Functional Derivatives functional derivatives have rules like regular derivatives derivative of a sum equals the sum of the derivatives product rule order of differentiation can usually be interchanged if F= F[f], but f = f[g], use the chain rule

40. Functional Derivatives why do we care about functional derivatives? • functional derivatives appear in the Kohn-Sham energy functional and Kohn-Sham operator Kohn-Sham energy functional: Kohn-Sham operator:

41. Functional Derivatives the energy operators in the Kohn-Sham operator are functional derivatives • consider the nuclear-electron attraction term • if we expand the functional in a Taylor series: This is called a local functional derivative because it only depends on the value of ρ at position r. The corresponding functional is called a local functional.

42. Functional Derivatives the energy operators in the Kohn-Sham operator are functional derivatives • consider the electron-electron Coulomb repulsion term • if we expand the functional in a Taylor series: This is called a non-local functional derivative because its value at r1 depends on ρ(r2). The corresponding functional is called a non-local functional.

43. Functional Derivatives the total energy functional contains functional derivatives

44. Functional Derivatives the energy operators in the Kohn-Sham operator are functional derivatives

45. Functional Derivatives why do we care about functional derivatives? • functional derivatives appear in the Kohn-Sham energy functional and Kohn-Sham operator • of particular importance if the exchange-correlation functional: • of course, we don’t have exact forms for these expressions • but we do know what they should account for

46. Toward XC Functionals what are the strategies used to develop functionals? Knowledge of constraints on the density and hXC • if we require the functional to meet certain constraints, we can focus on a limited (but still very large) set of forms • believe it or not, many successful functionals do not meet the most basic of constraints Mathematical and physical intuition • if we try to develop functionals that describe simple model systems, we may arrive at good functionals for real molecules and materials • we also have to select functional forms that satisfy the necessary constraints on the wavefunction Parameter selection • non-empirical: select parameters to meet constraints • empirical: select parameters to reproduce experimental/ab initio data