11-9 The Hartree–Fock Limit
Section 11-10 Correlation Energy
• aug-cc-pVnZ [cc-pVnZ basis sets augmented with one set of diffuse primitive gaussian functions for each angular momentum symmetry present in the cc-pVnZ basis set, to provide an accurate description of anions and weak interactions, e.g., van der Waals forces and hydrogen bonding.]
11-9 The Hartree–Fock Limit
It should be apparent that different choices of basis set will produce different SCF wavefunctions and energies. Suppose that we do an SCF calculation on some molecule, using a minimal basis set and obtain a total electronic energy E1. If we now choose a double-ζ basis and do a new SCF calculation, we will obtain an energy E2 that normally will be lower than E1. (If one happens to choose the first basis wisely and the second unwisely, it is possible to find E2 higher than E1. We assume here that each improvement to the basis extends the mathematical flexibility while including the capabilities of all preceding bases.) If we now add polarization functions and repeat the SCF procedure, we will find E3 to be lower than E2. We can continue in this way, adding new functions in bonds and elsewhere, always increasing the capabilities of our basis set, but always requiring that the basis describe MOs in a single determinantal wavefunction. The electronic energy will decrease with each basis set improvement, but eventually this decrease will become very slight for any improvement; that is, the energy will approach a limiting value as the basis set approaches mathematical completeness.
This limiting energy value is the lowest that can be achieved for a single determinantal wavefunction. It is called the Hartree–Fock energy. The MOs that correspond to this limit are called Hartree–Fock orbitals (HF orbitals), and the determinant is called the HF wavefunction.
Sometimes the term restricted Hartree–Fock (RHF) is used to emphasize that the wavefunction is restricted to be a single determinantal function for a configuration wherein electrons of α spin occupy the same space orbitals as do the electrons of β spin. When this restriction is relaxed, and different orbitals are allowed for electrons with different spins, we have an unrestricted Hartree–Fock (UHF) calculation. This refinement is most likely to be important when the numbers of α- and β-spin electrons differ. We encountered this concept in Section 8-13, where we noted that the unpaired electron in a radical causes spin polarization of other electrons, possibly leading to negative spin density.
11-10 Correlation Energy The Hartree–Fock energy is not as low as the true energy of the system. The mathematical reason for this is that our requirement that ψ be a single determinant is restrictive and we can introduce additional mathematical flexibility by allowing ψ to contain many determinants. Such additional flexibility leads to further energy lowering.
There is a corresponding physical reason for the HF energy being too high. It is con nected with the independence of the electrons in a single determinantal wavefunction.
To understand this, consider the four-electron wavefunction
ψ = φ
1(1)φ1(2)φ2(3)φ2(4)
(11-22)
• aug-cc-pVnZ [cc-pVnZ basis sets augmented with one set of diffuse primitive gaussian functions for each angular momentum symmetry present in the cc-pVnZ basis set, to provide an accurate description of anions and weak interactions, e.g., van der Waals forces and hydrogen bonding.]
11-9 The Hartree–Fock Limit
It should be apparent that different choices of basis set will produce different SCF wavefunctions and energies. Suppose that we do an SCF calculation on some molecule, using a minimal basis set and obtain a total electronic energy E1. If we now choose a double-ζ basis and do a new SCF calculation, we will obtain an energy E2 that normally will be lower than E1. (If one happens to choose the first basis wisely and the second unwisely, it is possible to find E2 higher than E1. We assume here that each improvement to the basis extends the mathematical flexibility while including the capabilities of all preceding bases.) If we now add polarization functions and repeat the SCF procedure, we will find E3 to be lower than E2. We can continue in this way, adding new functions in bonds and elsewhere, always increasing the capabilities of our basis set, but always requiring that the basis describe MOs in a single determinantal wavefunction. The electronic energy will decrease with each basis set improvement, but eventually this decrease will become very slight for any improvement; that is, the energy will approach a limiting value as the basis set approaches mathematical completeness.
This limiting energy value is the lowest that can be achieved for a single determinantal wavefunction. It is called the Hartree–Fock energy. The MOs that correspond to this limit are called Hartree–Fock orbitals (HF orbitals), and the determinant is called the HF wavefunction.
Sometimes the term restricted Hartree–Fock (RHF) is used to emphasize that the wavefunction is restricted to be a single determinantal function for a configuration wherein electrons of α spin occupy the same space orbitals as do the electrons of β spin. When this restriction is relaxed, and different orbitals are allowed for electrons with different spins, we have an unrestricted Hartree–Fock (UHF) calculation. This refinement is most likely to be important when the numbers of α- and β-spin electrons differ. We encountered this concept in Section 8-13, where we noted that the unpaired electron in a radical causes spin polarization of other electrons, possibly leading to negative spin density.
11-10 Correlation Energy The Hartree–Fock energy is not as low as the true energy of the system. The mathematical reason for this is that our requirement that ψ be a single determinant is restrictive and we can introduce additional mathematical flexibility by allowing ψ to contain many determinants. Such additional flexibility leads to further energy lowering.
There is a corresponding physical reason for the HF energy being too high. It is con nected with the independence of the electrons in a single determinantal wavefunction.
To understand this, consider the four-electron wavefunction
ψ = φ
1(1)φ1(2)φ2(3)φ2(4)
(11-22)