While HF theory was groundbreaking in its handling of the many-body problem, there still exists a large computational cost associated with building the many-electron wavefunction and solving the eigenvalue problem for large systems.31 The dimension of the wavefunction is directly related to the number of electrons (three-dimensions per electron), and for example the full wavefunction of the relatively simple CO2 molecule has 66-dimensions. Because the electron
wavefunction is not directly observable, an alternative method of solving the Schrödinger equation would be to shift the focus away from the wavefunction and instead use something that can be observed, specifically the electron density (𝑛) that has an operator,
𝑛̂ = 𝛿(𝑟 − 𝑟′) (2.41)
with the expectation value of the density given by,
𝑛(𝑟) = ⟨𝛹|𝑛̂|𝛹⟩ = ∫|𝛹(𝑟)|2𝑑𝑟 (2.42)
The electron density can be used in place of many of the wavefunction terms in the total energy Hamiltonian, 𝐻̂ = 𝑇̂ + 𝑍̂ + 𝑈̂ (2.43) 𝐻̂ = ∑ − ℏ 2𝑚𝑖∇𝑖2 𝑁 𝑖 + 𝜉 (∑ 𝒁𝐼 |𝑟𝑖− 𝑅𝐼|+ 1 2∑ ∑ 1 |𝑟𝑖 − 𝑟𝑗| 𝑁 𝑗≠𝑖 𝑁 𝑖 𝑁 𝑖,𝐼 ) (2.44)
55 For example, the electron-nuclei potential energy 𝑍, can be determined from the expectation value of the electron-nuclei potential energy operator 𝑍̂, given by
⟨𝛹(𝑟)|𝑍̂|𝛹(𝑟)⟩ = 〈𝑍̂〉𝛹(𝑟) = 𝜉 ∑ ∑ ∫ 𝜓𝑖(𝑟) 𝒁𝐼 |𝑟 − 𝑅𝐼| 𝑁𝑛 𝐼 𝑁𝑒 𝑖 𝜓𝑖(𝑟)𝑑𝑟 (2.45) 〈𝑍̂〉𝛹(𝑟) = 𝜉 ∑ ∑ ∫|𝜓𝑖(𝑟)|2 𝒁𝐼 |𝑟 − 𝑅𝐼| 𝑁𝑛 𝐼 𝑁𝑒 𝑖 𝑑𝑟 (2.46)
Substituting the expression for electron density, the equation becomes
〈𝑍̂〉𝑛(𝑟) = 𝜉 ∑ ∫ 𝑛(𝑟) 𝒁𝐼 |𝑟 − 𝑅𝐼|𝑑𝑟 𝑁𝑛 𝐼 (2.47)
Expressing the energy of a system in terms of electron density rather than the many-body wavefunction is good in theory, but without knowledge of the exact wavefunction determining
n(r) is still an impossible task. However in 1964 Pierre Hohenberg and Walter Kohn developed a
theory that claimed the ground state energy for a many-body wavefunction can be found by solving the non-interacting electron equations in an effective potential determined solely from the electron density.23 Hohenberg and Kohn supported this by introducing two important theorems: (1) that there cannot be two different external potentials that yield the same non- degenerate ground state charge density, and (2) that the external potential (and therefore the total energy), is a unique functional of the electron density.
The first theorem can be proven using a reductio ad absurdum argument. Assuming that there exist two different external potentials, 𝑉𝑒𝑥𝑡(1)(𝑟) and 𝑉𝑒𝑥𝑡(2)(𝑟) that yield the same ground state
56 electron density, 𝑛(𝑟), the potentials would be a result of two distinct Hamiltonians, 𝐻̂(1) and
𝐻̂(2), which give way to unique wavefunctions 𝛹(1)(𝑟) and 𝛹(2)(𝑟). The variational principle
states that that no wavefunction can yield an energy for 𝛹(1)(𝑟) less than the expectation value
using 𝐻̂(1)(𝑟),
𝐸(1) = ⟨𝛹(1)|𝐻̂(1)|𝛹(1)⟩ < ⟨𝛹(2)|𝐻̂(1)|𝛹(2)⟩ (2.48)
Because of the initial assumption that states the two systems have identical densities, the second expectation value in the above expression can be rewritten as
⟨𝛹(2)|𝐻̂(1)|𝛹(2)⟩ = ⟨𝛹(2)|𝐻̂(2)|𝛹(2)⟩ + ∫ 𝑛(𝑟) (𝑉̂
𝑒𝑥𝑡(1)(𝑟) − 𝑉̂𝑒𝑥𝑡(2)(𝑟)) 𝑑𝑟 (2.49)
Switching the labels, it is equally shown that the following expression is valid,
⟨𝛹(1)|𝐻̂(2)|𝛹(1)⟩ = ⟨𝛹(1)|𝐻̂(1)|𝛹(1)⟩ + ∫ 𝑛(𝑟) (𝑉̂
𝑒𝑥𝑡(2)(𝑟) − 𝑉̂𝑒𝑥𝑡(1)(𝑟)) 𝑑𝑟 (2.50)
And by combining (2.46) and (2.47) the relationship,
𝐸(1)+ 𝐸(2)< 𝐸(2)+ 𝐸(1) (2.51)
is obtained, which is clearly contradictory. This contradiction proves that each ground state electron density does in fact determine a unique external potential.
The second theorem provides a means of solving for the energy of a system based solely on the electron density, by stating that a unique functional for the energy can be defined using
57 the density and that the exact ground state energy is the minimum value of the functional. This uses the previous theorem, that because the external potential and therefore the wavefunction is determined uniquely by the density then all other observables must also be uniquely determined.
𝐸̂[𝑛(𝑟)] = 𝑇̂[𝑛(𝑟)] + 𝑍̂[𝑛(𝑟)] + 𝑉̂𝑒𝑥𝑡[𝑛(𝑟)] (2.52)
𝐸̂[𝑛(𝑟)] ≡ 𝐹̂[𝑛(𝑟)] + 𝑉̂𝑒𝑥𝑡[𝑛(𝑟)] (2.53)
Here, 𝐹̂[𝑛(𝑟)] is a universal functional (not to be confused with the HF Fock operator) because
its treatment of the kinetic and internal potential energies is the same for all systems, that is it does not depend on 𝑉𝑒𝑥𝑡. Again, introducing the variational principle, it follows that
minimization of 𝐸[𝑛(𝑟)] with respect to 𝑛(𝑟) will result in the lowest ground state energy,
𝐸(1)= 𝐸[𝑛(𝑟)] = ⟨𝛹(1)|𝐻̂(1)|𝛹(1)⟩ < ⟨𝛹(2)|𝐻̂(1)|𝛹(2)⟩ = 𝐸(2) (2.54)
These two theorems, while they showcase that the electron density can be used to effectively calculate the ground state properties of a given system, do not provide any methods for actually solving the problem. The issues arise from the fact that 𝐹̂[𝑛(𝑟)] is not known, and
58 A year after Hohenberg and Kohn published their theorems, Walter Kohn and Lu Jeu Sham developed an ansatz which stated that the exact ground state density can be written as the sum of non-interacting particle wavefunctions,
𝑛(𝑟) = ∑|𝜓𝑖(𝑟)|2 𝑁
𝑖
(2.55)
enabling the use of an independent particle equation that can be solved numerically.24
The method used in Kohn-Sham density functional theory (KS-DFT) is very similar to what is done in HF, with a few notable exceptions.39 First, the KS-energy is constructed in a slightly different manner, as (unlike HF) electron exchange is not taken into account. The energy can be written as,
𝐸[𝑛(𝑟)] = 𝑇[𝑛(𝑟)] + 𝑉[𝑛(𝑟)] + 𝑈[𝑛(𝑟)]
= 𝑇𝑠[𝑛(𝑟)] + 𝑍[𝑛(𝑟)] + 𝑈𝐻[𝑛(𝑟)] + 𝐸𝑥𝑐[𝑛(𝑟)] (2.56) The notable differences include referring to the kinetic and electron-electron repulsion energy terms as 𝑇𝑠 and 𝑈𝐻, respectively, and the addition of the exchange-correlation energy term, 𝐸𝑥𝑐.
In KS-DFT, the kinetic energy term is split into two contributions, the non-interacting kinetic energy (𝑇𝑠) and the kinetic energy that arises due to correlation effects (𝑇𝑐), such that,
59 The correlation component is included in the exchange-correlation energy (vide infra). The repulsion energy in KS-DFT is referred to as the Hartree potential, since it is simple the classical electrostatic interaction between the density with itself,
𝑈𝐻[𝑛(𝑟)] = 1 2𝜉 ∫ ∫
𝑛(𝑟)𝑛(𝑟′)
|𝑟 − 𝑟′| 𝑑𝑟𝑑𝑟′ (2.58) Finally, all of the missing energy factors, including electron exchange, correlation, and 𝑇𝑐 are
grouped into 𝐸𝑥𝑐. There exist various approximations for 𝐸𝑥𝑐, many of which are discussed in
detail in Chapter 3.
The KS equation is a simple minimization of the total energy with respect to the KS- orbitals, in order to obtain the orbitals22,
𝛿𝐸𝐾𝑆 𝛿𝜓𝑖(𝑟)= ( 𝛿𝑇𝑠 𝛿𝜓𝑖(𝑟)+ [ 𝛿𝑉 𝛿𝑛(𝑟)+ 𝛿𝐸𝐻𝑎𝑟𝑡𝑟𝑒𝑒 𝛿𝑛(𝑟) + 𝛿𝐸𝑥𝑐 𝛿𝑛(𝑟)]) 𝛿𝑛(𝑟) 𝛿𝜓𝑖(𝑟)= 𝐸𝑖(𝑟)𝜓𝑖(𝑟) (2.59) which simplifies to 𝛿𝐸𝐾𝑆 𝛿𝜓𝑖(𝑟)= − ℏ 2𝑚∇𝑖2𝜓𝑖(𝑟) + [𝜉 ∫ 𝜓𝑖(𝑟) 𝒁𝐼 |𝑟 − 𝑅𝐼|𝑑𝑟 + 1 2𝜉 ∫ ∫ 𝜓𝑖(𝑟) 𝑛(𝑟′) |𝑟 − 𝑟′|𝑑𝑟𝑑𝑟′+ 𝛿𝐸𝑥𝑐 𝛿𝑛(𝑟)] 𝜓𝑖(𝑟) (2.60)
Where the bracketed term is typically represented by 𝑣𝑠(𝑟). Therefore, the electron density is
what is used to determine the one-electron orbitals, using the same method as the Hartree-Fock- Roothaan equation (2.39), specifically using an iterative SCF technique for determining the orbital coefficients (Figure 2-1).
60 There are many similarities between HF and KS-DFT, with only minor differences in the manipulation of the mathematics governing the two techniques. Arguably the most impactful dissimilarity is the explicit exchange and lack of correlation energy in HF and the inclusion of the approximate exch22ange-correlation energy in KS-DFT.20 The next chapter details the various formulations of exchange-correlation functional, as well as describing how the atomic orbitals are modeled, which is applicable to both methods.
61
2.3 References
1. Selleri, F., Wave-Particle Duality; Springer, 1992.
2. Heisenberg, W., Physics and Philosophy: The Revolution in Modern Science. 1958. 3. Müller-Kirsten, H. J., Introduction to Quantum Mechanics: Schrödinger Equation and
Path Integral; World Scientific, 2006.
4. Einstein, A., A Heuristic Viewpoint Concerning the Production and Transformation of
Light, 1929.
5. De Broglie, L., The Reinterpretation of Wave Mechanics. Foundations of Physics 1970,
1, 5-15.
6. Green, H. S.; Born, M., Matrix Mechanics. 1965.
7. Heisenberg, W., Über Quantentheoretische Umdeutung Kinematischer Und
Mechanischer Beziehungen; Springer, 1985.
8. Waerden, B. L., Sources of Quantum Mechanics; Courier Corporation, 1968; Vol. 5. 9. Schrödinger, E., An Undulatory Theory of the Mechanics of Atoms and Molecules. Phys.
Rev. 1926, 28, 1049.
10. Phillips, A. C., Introduction to Quantum Mechanics; John Wiley & Sons, 2013. 11. Leforestier, C.; Bisseling, R.; Cerjan, C.; Feit, M.; Friesner, R.; Guldberg, A.;
Hammerich, A.; Jolicard, G.; Karrlein, W.; Meyer, H.-D., A Comparison of Different Propagation Schemes for the Time Dependent Schrödinger Equation. J. Comput. Phys.
1991, 94, 59-80.
12. Cohen, L., Hamiltonian Operators Via Feynman Path Integrals. Journal of Mathematical
Physics 1970, 11, 3296-3297.
62 14. Griffiths, D. J.; Harris, E. G., Introduction to Quantum Mechanics. Am. J. Phys. 1995, 63,
767-768.
15. Duncan, A., Rydberg Series in Atoms and Molecules; Elsevier, 2012; Vol. 23.
16. Eckert, K.; Schliemann, J.; Bruss, D.; Lewenstein, M., Quantum Correlations in Systems of Indistinguishable Particles. Annals of Physics 2002, 299, 88-127.
17. Zanardi, P., Quantum Entanglement in Fermionic Lattices. Phys. Rev. A 2002, 65, 042101.
18. Murdin, P., Pauli Exclusion Principle. Encyclopedia of Astronomy and Astrophysics
2000, 1, 4896.
19. Kittel, C., Introduction to Solid State Physics; Wiley, 2005.
20. Levine, I. N., Quantum Chemistry; Pearson Prentice Hall Upper Saddle River, NJ, 2009; Vol. 6.
21. Slater, J. C., A Simplification of the Hartree-Fock Method. Phys. Rev. 1951, 81, 385-390. 22. Evarestov, R. A., Quantum Chemistry of Solids. 2012.
23. Hohenberg, P.; Kohn, W., Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864- B871.
24. Kohn, W.; Sham, L. J., Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133-A1138.
25. Kohn, W.; Becke, A. D.; Parr, R. G., Density Functional Theory of Electronic Structure.
J. Phys. Chem 1996, 100, 12974-12980.
26. Bersuker, I. B., Electron Structure and Properties of Coordination Compounds- Introduction to the Theory. Leningrad Izdatel Khimiia 1976, 1.
63 27. Harrison, W. A., Electronic Structure and the Properties of Solids: The Physics of the
Chemical Bond; Courier Corporation, 2012.
28. Born, M., Born-Oppenheimer Approximation. Ann. Physik 1927, 84, 457.
29. Alexander, M. H.; Capecchi, G.; Werner, H.-J., Theoretical Study of the Validity of the Born-Oppenheimer Approximation in the Cl+ H2→ Hcl+ H Reaction. Science 2002, 296, 715-718.
30. Ekeland, I., On the Variational Principle. J. Math. Anal. Appl. 1974, 47, 324-353. 31. Sholl, D.; Steckel, J. A., Density Functional Theory: A Practical Introduction; John
Wiley & Sons, 2011.
32. Csizmadia, I. G., Theory and Practice of Mo Calculations on Organic Molecules. . Int. J.
Quantum Chem 1978, 13, 159-159.
33. Gritsenko, O.; Schipper, P.; Baerends, E., Exchange and Correlation Energy in Density Functional Theory: Comparison of Accurate Density Functional Theory Quantities with Traditional Hartree–Fock Based Ones and Generalized Gradient Approximations for the Molecules Li2, N2, F2. J. Chem. Phys. 1997, 107, 5007-5015.
34. Lykos, P.; Pratt, G., Discussion on the Hartree-Fock Approximation. Reviews of Modern
Physics 1963, 35, 496.
35. Kerman, A.; Svenne, J.; Villars, F., Hartree-Fock Calculation for Finite Nuclei with a Nonlocal Two-Body Potential. Phys. Rev. 1966, 147, 710.
36. Langhoff, P.; Karplus, M.; Hurst, R., Approximations to Hartree—Fock Perturbation Theory. J. Chem. Phys. 1966, 44, 505-514.
64 37. Clementi, E.; Roetti, C., Roothaan-Hartree-Fock Atomic Wavefunctions: Basis Functions
and Their Coefficients for Ground and Certain Excited States of Neutral and Ionized Atoms, Z≤ 54. At. Data Nucl. Data Tables 1974, 14, 177-478.
38. McQuarrie, D. A., Quantum Chemistry; University Science Books, 2008.
39. Bickelhaupt, F. M.; Baerends, E. J., Kohn‐Sham Density Functional Theory: Predicting and Understanding Chemistry. Reviews in Computational Chemistry, Volume 15 2007, 1- 86.
65
CHAPTER 3: Density Functional Theory