Performance of Hartree-Fock and Correlated Methods

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1 Chemistry 460 Fall 2017 Dr. Jean M. Standard December 4, 2017 Performance of Hartree-Fock and Correlated Methods Hartree-Fock Methods Hartree-Fock methods generally yield optimized geomtries and molecular properties that are in good agreement with experimental results, though there are some exceptions. General rules of thumb for Hartree-Fock calculations employing a double zeta plus polarization (DZP) basis set are as follows: - Bond lengths and bond angles usually are predicted to within about 1 percent of experimental values. - Conformational energies usually are predicted to within 1-2 kcal/mol of the experimental results. - Vibrational frequencies usually are too high by about 10 percent due to anharmonicity effects (low frequency modes are often the furthest off). Tables 1 and 2 show results from Hartree-Fock calculations employing various basis sets to determine the equilibrium geometries of simple molecules. The basis set size increases from the first one listed to the last one listed. While there is not a monotonic change in bond length from the smallest basis set to the largest basis set, in most cases the largest basis set will produce the closest agreement with experiment. Table 1.* Bond lengths of simple first-row molecules from Hartree-Fock calculations Bond Lengths (a.u.)** Basis set CH 4 NH 3 H 2 O HF STO-3G G G* G** Experiment * Tables 1 and 2 are adapted from J. B. Foresman and A. Frisch, Exploring Chemistry with Electronic Structure Methods, Second Ed., Gaussian, Inc., Pittsburgh, PA, ** Note that 1 a.u. of length equals Å. Table 2. Bond angles of simple first-row molecules from Hartree-Fock calculations Bond Angle (degrees) Basis set NH 3 H 2 O STO-3G G G* G** Experiment Hartree-Fock calculations also provide excellent agreement with experiment for a variety of conformational and thermochemical properties as discussed in class. One of the cases where Hartree-Fock calculations fail is in the calculation of dissociation energies. This is because these calculations usually start with a molecule that has a closed shell of electrons and the molecule breaks into two fragments that are open shell molecules. A closed shell molecule has all its electrons paired; in an open shell molecule, there is at least one unpaired electron. The energy difference between closed and open shell molecules is very difficult to calculate using Hartree-Fock methods.

2 Table 3 gives the results of molecular orbital calculations of the dissociation of the HF molecule into H and F atoms. The Hartree-Fock result is very far off from the experimental dissociation energy. Methods that include electron correlation, such as the others listed in the table (MP2, MP4, QCISD), are required in order to properly treat the dissociation energy. 2 Table 3. Dissociation energies from molecular orbital calculations for the dissociation of the HF molecule into H and F atoms. Method Dissociation Energy (kcal/mol) Hartree-Fock 97.9 MP MP QCISD Experiment As another example of the poor behavior of the Hartree-Fock method for mapping potential energy curves, Figure 1 shows the energy of OH as a function of the O-H bond distance for the Hartree-Fock method and for a higher level method (CI, or Configuration Interaction). These computational methods are compared with the experimental curve. Figure 1. Calculated and experimental potential energy curves for OH. Electron Correlation: Definition In Hartree-Fock theory, the Fock operator depends on the coordinates of only one electron. It turns out therefore that when the Hartree-Fock equations are solved self-consistently, it is as if the equations were being solved for the motion of a single electron moving in the average field of all the others. This procedure neglects the point that electrons do not want to be close to one another due to repulsion. Rather, the electrons are correlated they want to stay away from each other. The Hartree-Fock energy therefore is higher than the true energy not only because of the Variation Principle but because of the inclusion of interactions that normally would not occur when more than one electron is treated simultaneously.

3 Because of the neglect of electron correlation, the Hartree-Fock limit that is approached as the basis set increases is still higher than the exact energy, as shown in Figure 2. 3 Eapprox H-F Limit Eexact No. of Basis Functions Figure 2. Behavior of the approximate electronic energy as a function of the basis set size. The electron correlation energy, E corr, is defined as the difference in energy between the exact energy and the H-F limit, E corr = E exact E HFlimit. (1) The correlation energy is a small fraction of the exact energy, usually ranging from 0.1 to 1.0% of the exact energy. However, to do the best job in calculating the wavefunction, geometry, and properties of a molecule, electron correlation must be included. Some specific values of the estimated energies of the Hartree-Fock limit and the correlation energy are shown in Table 4. Table 4. Hartree-Fock limit energies and electron correlation energies for a selection of atoms and small molecules listed in order of decreasing energy. Note that 1 a.u. = kcal/mol. Atom or Molecule No. of Electrons E(HF limit) (a.u.) E(correlation) (a.u.) H He BH O CH NH H 2 O HF N CO Ne Si S Ar %

4 An example of approaching the Hartree-Fock limit for the energy with respect to increasing basis set size is shown in Figure 3 for calculations on the H 2 molecule. Also shown are two different methods, discussed below, for the inclusion of electron correlation, MP2 and CCSD. 4 Figure 3. Energy of H 2 as a function of increasing basis set size The Hartree-Fock calculations of H 2 after basis functions are near the Hartree-Fock limit of a.u. The MP2 calculation, which is the most common method for inclusion of electron correlation, recovers a.u. of the correlation energy, about 69% of the total correlation energy. The CCSD calculation recovers a.u. of the correlation energy, or about 87% of the total. The geometry also converges as the basis set increases but not necessarily in a monotonic fashion. Shown in Figure 4 is the optimized H-H bond distance for Hartree-Fock and correlated calculations as the number of basis functions increases. Figure 4. Bond distance of H 2 as a function of increasing basis set size

5 5 Electron Correlation Methods There are many ways to include electron correlation in a molecular orbital calculation. The most common method is called MØller-Plesset Perturbation Theory (or MP2, MP4, etc.). This method first involves a Hartree-Fock calculation, and then the electron correlation is treated as a perturbation to the Hartree-Fock solution. This allows for the correlation of the electrons to be added to the Hartree-Fock solution as a small correction. The most common Moller-Plesset Perturbation Theory is called MP2 (for second-order perturbation theory). More accurate perturbation methods such as fourth-order (MP4) can be used, though the difficulty and expense of these methods increase rapidly. The MP2 method usually recovers a significant proportion of the correlation energy, so it is usually sufficient (and cost effective) for obtaining fairly high accuracy results. Shown in Tables 5 and 6 are bond lengths and angles of small molecules determined using the MP2 method. Hartree-Fock results for the same molecules were shown in Tables 1 and 2. Table 5.* Bond Lengths of Simple First-Row Molecules from MP2 Calculations Bond Length (a.u.) Basis set CH 4 NH 3 H 2 O HF STO-3G G G* G** Experiment * Tables 5 and 6 are adapted from J. B. Foresman and A. Frisch, Exploring Chemistry with Electronic Structure Methods, Second Ed., Gaussian, Inc., Pittsburgh, PA, Table 6. Bond Angles of Simple First-Row Molecules from MP2 Calculations Bond Angle (degrees) Basis set NH 3 H 2 O STO-3G G G* G** Experiment Note that while the Hartree-Fock results are acceptable, to get truly good agreement with experiment, electron correlation is required. The MP2 results are in general much closer to experiment than the Hartree-Fock results, especially for the more polar molecules. However, even for the MP2 method, a basis set of a reasonable size must be used. Notice that the minimal basis set STO-3G does poorly even with the MP2 method. Other electron correlation methods include Configuration Interaction (CI) and variants such as Quadratic Configuation Interaction (QCI). These methods treat the electron correlation in a completely different way than the MP2 method. Rather than using a single wavefunction that represents a particular occupancy of molecular orbitals, CI uses a linear combination of wavefunctions, each of which represents a different molecular orbital occupancy. This has the effect of treating electron correlation by allowing the electrons to spread out more. Unfortunately, CI is a very computationally demanding method, and usually some approximation to the full set of combinations of different wavefunctions is used. These methods have names like CISD for Configuration Interaction with Single and Double excitations or QCISD(T) for Quadratic Configuration Interaction with Single, Double, and selected Triple excitations.

6 Related correlated techniques include those that are referred to as coupled cluster methods, such as Coupled Cluster with Single and Double excitations (and selected Triple excitations), CCSD(T). This method is currently thought to be one of the best for obtaining highly accurate geometries and vibrational frequencies. 6 Another method that treats electron correlation is called the Multi-Configuration Self-Consistent Field method, or MCSCF. A related method is the Complete Active Space Self-Consistent Field method, or CASSCF. These methods are particularly useful for systems that are biradicals or have some biradical character. One example of a type of molecular system where the MCSCF method is appropriate is for triplet carbenes (and even some singlet carbenes). In the MCSCF method, the wavefunction used is similar to that for the Configuration Interaction method, except that not only are the linear coefficients of the different wavefunctions allowed to vary, but so are the orbitals which comprise the wavefuctions. Once again, this makes the MCSCF method rather computationally demanding. CPU Times The CPU time required for a Hartree-Fock calculation grows rapidly as the size of the molecule and the number of basis functions increases. It is estimated that for a single-point energy calculation using the Hartree-Fock method that the CPU time required scales roughly as the number of basis functions to the power 3.5 (that is, as K 3.5, where K is the number of basis functions). For higher-level calculations that include electron correlation, the CPU time scales even more steeply with the number of basis functions. For MP2 calculations, the CPU time scales roughly as OK 4, where O is the number of occupied molecular orbitals and K is the number of basis functions. For MP4 and QCISD(T), the CPU time scales approximately as O 3 V 4, where O is the number of occupied molecular orbitals and V is the number of virtual (or empty) molecular orbitals. Some actual relative CPU times for Hartree-Fock calculations are shown in Tables 7 and 8. Table 7.* Relative CPU Time for Single Point Energy Calculation on CH 4 Basis Set (Number of Basis Functions) Method STO-3G (9) 3-21G (17) 6-31G* (23) 6-31+G* (27) G(2d,p) (55) HF MP MP QCISD(T) * Tables 7 and 8 are adapted from J. B. Foresman and A. Frisch, Exploring Chemistry with Electronic Structure Methods, Second Ed., Gaussian, Inc., Pittsburgh, PA, Table 8. Relative CPU Time for Single Point Energy Calculation on C 5 H 12 Basis Set (Number of Basis Functions) Method 3-21G (69) 6-31G* (99) 6-31+G* (119) G(2d,p) (219) HF MP MP QCISD(T)

7 It is clear from these tables that the methods which include electron correlation are much more demanding. The effect is even larger for medium to large molecules. Figure 5 provides a graphical illustration of the CPU time required for calculations at various levels of theory and different basis set sizes. Notice that the scale on the z-axis is logarithmic. 7 Figure 5. Relative CPU times for various levels of calculation and basis sets [from J. B. Foresman and A. Frisch, Exploring Chemistry with Electronic Structure Methods, Second Ed., Gaussian, Inc., Pittsburgh, PA, 1996].