Chemistry 380.37 Fall 2015 Dr. Jean M. Standard October 28, 2015 Density Functional Theory What is a Functional? A functional is a general mathematical quantity that represents a rule to convert a function into a number. This should not be confused with a function, which represents a rule for converting one number into another number. The rule for a typical functional might be given as a definite integral. For example, the rule for a particular functional might be to integrate a function on the interval from 0 to 1. If the function is f x applying the rule is ( ) = x 3, then the result of F[ f ( x) ] = f ( x) dx 1 = x 3 dx = 0 F[ f ( x) ] = 1 4. 1 0 x 4 1 4 0 [ ( )]. This notation means that F is the functional that ( ). The rule in this case is to take the definite integral of the function to produce a Note that a functional is represented by square brackets, F f x works on the function f x number. The rule for any functional must be specified in order to know how to produce a number from any given function. The Hohenberg-Kohn Theorems Two scientists, Walter Kohn and Pierre Hohenberg, developed a pair of theorems in 1964 related to the electron density. Unlike the quantum mechanical wavefunction, Ψ( x 1, y 1,z 1, x 2, y 2,z 2, x N, y N,z N ), which is not physically observable and depends upon the coordinates of all N electrons of a system, the electron density ρ x, y,z is physically observable and depends upon only three spatial coordinates. Technically, the electron density can be obtained from the electronic wavefunction using the relation ( ) ( ) = dr 2 d r 3 ρ x, y,z dr N Ψ * ( r 1,r 2, r N ) Ψ( r 1,r 2, r N ). (1) Notice that the integration is over all the electron coordinates except one set. The electron density ρ( x, y,z) therefore is the probability density for finding an electron at the point ( x, y,z). The choice to integrate over all the electron coordinates except those of electron 1 is arbitrary. Since the electrons are indistinguishable, the expression above represents the probability density for finding any one electron at the point ( x, y,z). The first Hohenberg-Kohn theorem states that the energy eigenvalue (i.e., the electronic energy of the system) is a functional of the electron density, E = F[ ρ( x, y,z) ]. (2) Thus, if we have the electron density and we know the specific functional that relates the density to the energy, we can determine exactly the energy eigenvalue. Note that this theorem technically applies to electronic ground states only, so it does not treat electronically excited states. This theorem tells us that a functional exists, but it does not tell us how to find it.
2 The second Hohenberg-Kohn theorem provides a Variation Principle for density functional theory. Since we do not know the exact electron density, we can place bounds on any approximate or trial electron density, ρ t x, y,z ( ), [ ] E ρ exact x, y,z E ρ t ( x, y,z) [ ( )]. (3) The bounding property only works for the exact functional, though. Since we do not know it, we have to come up with an empirical functional and an approximate electron density. A variety of empirical functionals are employed in density functional theory (DFT) calculations today. Some common functionals are, for example, B3LYP, B3PW91, B3P86, BLYP, BP86, CAM-B3LYP, PBE, M062X, and so on. Kohn-Sham Equation and Kohn-Sham Orbitals In order to obtain an electron density from DFT, some assumptions are made that are analogous to those made in solving the Schrödinger equation. For the Schrödinger equation, we employed the LCAO-MO approximation to obtain a trial wavefunction. In DFT, we employ a similar approximation for the trial electron density, ( ) = φ KS i ( x, y,z) 2 ρ t x, y,z. (4) i KS Here, φ i ( x, y,z ) are Kohn-Sham orbitals. These are similar to the molecular orbitals that appear in our approach for solving the Schrödinger equation approximately using the Hartree-Fock method. Each Kohn-Sham orbital holds two electrons and can be represented using a linear combination of atomic orbital basis functions, ( ) = c µi f µ ( x, y,z) φ i KS x, y,z K. (5) µ=1 In this equation, c µi correspond to numerical coefficients and the functions f µ ( x, y,z) are atomic orbital basis functions. The coefficients c µi are obtained by solving a set of equations called the Kohn-Sham equations. These are analogous to the Hartree-Fock equations employed when obtaining a solution of the Schrödinger equation. Advantages and Disadvantages of DFT One advantage of DFT is that unlike the Hartree-Fock method, DFT includes some electron correlation. The accuracy of DFT methods tends to be between that of Hartree-Fock and MP2. Another advantage is that DFT methods are cost efficient. The CPU time for DFT calculations tends to be similar to that of Hartree-Fock calculations. Thus, some electron correlation can be included using DFT for much less cost than it takes to carry out an MP2 calculation. One disadvantage of DFT is that there are many functionals available. It is not always clear what the best choice of functional is for the system of interest. Many DFT methods perform poorly for hydrogen-bonded systems and other weakly-bound intermolecular complexes, so care must be taken in selecting the DFT functional. Another disadvantage of DFT is that it cannot be systematically improved. With the Hartree-Fock method, we know that we can systematically improve the results by carrying out an MP2 or MP4 calculation, for example. With DFT, there is only one level of theory. The only improvements that can be made are in the basis set.
3 Examples of DFT Calculations 1. Geometries Shown in Figure 1 are some example calculations of simple organic molecules comparing geometries obtained from DFT to MP2 and experiment. DFT tends to perform well for typical organic molecules. In general, it also performs much better than Hartree-Fock for transition metal complexes and organometallic systems. Figure 1. Illustrative DFT results for geometries of simple organic molecules [from Computational Chemistry, E. Lewars, Kluwer Academic Publishers, 2003, p. 401.]
4 2. Example Reaction Energies and Transition State Geometries DFT results are reasonably good for energies of reaction, and generally the results are improved over the Hartree- Fock method. Figure 2 shows geometries of reactants, transition states, and products for a few example reactions. Figure 2. DFT results for geometries and energies (in kj/mol) of some example reactions [from Computational Chemistry, E. Lewars, Kluwer Academic Publishers, 2003, p. 405.] Notice that DFT does better with the geometries of stable structures than it does with transition state structures. In addition, DFT performs reasonably well in predicting energies of reaction and as well as energies of activation for these examples. In other cases, DFT does not perform as well for transition states and activation energies, and care must be taken in the choice of functional employed.
5 3. Bond Dissociation Energies Not only does DFT generally perform well for geometries, it also does a reasonably good job for energies because it includes some electron correlation. As a result, DFT performs well in even cases where the Hartree-Fock method performs poorly, such as for bond dissociation reactions. As an example, consider the dissociation of ethane, H 3 C-CH 3 2 CH 3. (6) Since the reaction leads to the formation of two radicals from a molecule in which all the electrons are paired, the electron correlation effects will be different on the product and reactant sides. Table 1 shows results from Hartree- Fock, MP2, and DFT calculations of the ethane dissociation energy. Table 1. Calculated dissociation energy for ethane at various levels of theory. Level/Basis Set Dissociation energy (kcal/mol) HF/6-31G(d) 59.3 MP2/6-31G(d) 88.4 B3LYP/6-31G(d) 86.8 Experiment 88.0-90.1 4. Excited States and UV-Vis Spectra An extension of the DFT method may be used to obtain electronically excited states and predict UV-Vis spectra. This method is known as Time-Dependent DFT (TDDFT) and is an efficient method for determining excited states compared to the high-level computationally expensive Configuration Interaction methods that previously were required for accurate results. Table 2. Calculated and experimental peaks (in nm) in the UV spectrum of methylenecyclopropane. TDDFT CIS/6-31+G(d) Experiment B3P86/6-311++G(d,p) 208 (100) 193 (100) 206 (100) 309 (26) 224 (15) 308 (13) Relative intensities are shown in parentheses.