Solving the One-‐Electron Problem in a Basis Set + Hartree-‐Fock In
Solving the one-electron problem in a basis set + Hartree-Fock in a nutshell.
May 16 2017, notes by Torin Hills
The easiest method of solving the one-electon problem is to introduce a set of basis functions. A basis allows us to solve the generalized Eigenvalue problem as a linear algebra problem.
Recall that our orbital energy eigenvalue problem is:
ℎ � �! = �!�!
! ! !" !" !! ℎ � = − ∇! + � (�) � � = ! ! !!!!
This differential equation has no analytic solution unless there are a (very) small number of atoms. To effectively solve this problem, we will introduce a fixed basis set: χλ(�). The basis that we choose is very important as it determines the precision of the calculated orbital energies. In practice, we find that increasing the size of the basis set gives better results.
Assuming {χλ(�)} is orthonormal and is a basis of N functions, we can write ϕ(�) as a linear combination of χλ(�) with linear expansion coefficients
!
� � ≈ �!(�)�! !!!
Our basis χλ(�) is fixed, so in order to optimize ϕ(�) we vary the value of �!. The variational principle is the best way to optimize ϕ(�) which gives the minimal orbital energy �.
The Variational Principle � ℎ � � ≥ � �
If we expand this with respect to our basis
∗ ! ∗ !,! �! ( � ��! � ℎ � �! � ) �! � ≥ ∗ ! ∗ !,! �! ( � ��!(�)�!(�))�!
! ∗ ! ∗ ℎ!" = � ��! � ℎ � �! � �!" = � ��!(�)�!(�)
The hλμ matrix is the one electron matrix, it gives the expectation value of the orbital energy with respect to the fixed basis.
The Sλμ matrix is the overlap integral. If the basis is orthonormal it would just be given by the unit matrix
(and Sλμ is the kronecker delta). With non-orthonormal functions the integral can be non-zero if the two basis functions are non-zero in the same region in space. So we can write our energy as
∗ !,! �! (ℎ!,!) �! � ≥ ∗ !,! �! (�!,!)�! If the overlap matrix is the unit matrix, the problem of minimizing the energy can be solved by diagonalizing the h matrix. The lowest eigenvalues is the variational solution and the corresponding eigenvector yields the optimal coeffcients C.The other eigenvalues and eigenvectors also represent approximate solutions to the one-electron problem.
�! = �!�!" !
We define ‘a’ as the eigenvector label (orbital) and ‘λ’ as the label for basis functions. By collecting the Cλ terms as column vectors we can represent all orbitals in the Cλa matrix.
�!! ⋯ �!! �!" = ⋮ ⋱ ⋮ �!" ⋯ �!" The lowest orbital energy is greater than or equal to the true lowest energy, as a staightforward application of the variational principle. However for the linear expansion problem there is a much more general principle. All eigenvalues are upper bounds to the true eigenvalues. Moreover, by increasing the size of the basis set (i.e. adding additional functions), every eigenvalue can only decrease (i.e. improve or stay the same). The excitation energy eigenvalues are all saddle points, as the energy can be lowered by mixing in lower energy eigenstates. However, such mixed states do not represent eigenstates, or stationary states of the Hamiltonian.
Another Derivation
Starting off again at the eigenvalue problem
ℎ�! = �!�!
�! = �!�!" !
ℎ �!�!" = �! �!�!" ! !
∗ Multiplying both sides by �!and integrating over all space
! ∗ ! ∗ � � �!ℎ�!�!" = � � �! �!�!"�! ! ! ! ∗ ∗ � � ( �!ℎ�!�!" − �! �!�!"�!) = 0 ! !
� ℎ � �!" − �!"�!"�! = 0 ! !
This condition is always true regardless of othonormality. If we would assume the condition of orthonormality, we would recover the previous equations:
If Sλν= δλν then SλνCνa=Cλa
ℎ!"�!" = �!"�! ∀� = 1, � !
�ℂ = ℂ�
For us to prove this condition we can use the definition for orthonormality and the definition of the overlap matrix to define the conditions required for �ℂ = ℂ�.
! ∗ 1 = � �( �!�!")( �!�!") ! !
∗ 1 = �!"�!"�!" !,!
† 1 = (� )!!�!"�!" !,!
ℂ†�ℂ = 1
Summary
! ∗ ℎ!" = � ��! � ℎ � �! �
! ∗ �!" = � ��!(�)�!(�)
Our general solution is �ℂ = �ℂ� with ℂ†�ℂ = 1 .
Basis Types
Suitable basis functions for chemistry must be ‘centered’ on the nuclei. The orbital that is occupied can be represented by different types of orbital functions. An orbital function has two components, the radial part and the angular part. The first type is the Slater Type Ortbital (STO):
!! !!!! ! � . �! (�!, �!)
Where Ra is the radius from the origin to the nuclei, θa and ϕa are the radial and azimuthal angles respectively. STOs can be used for very precise calculations for an electron, they are the most accurate representation we have of many orbitals however the integrals are difficult and expensive to calculate for the one-electron problems and become more complex for the two-electron integrals.
The second type of orbital is the Gaussian Type Orbital (GTO):
! !! !!!! ! � . �! (�!, �!)
Gaussian orbitals are much easier to calculate in both the one and two electron terms but give slightly less precise results as compared to the Slater type. Typically one use linear combinations of different Gaussians with different exponents, and the same angular part. These are called contracted gaussian basis functions. The integrals are calculated for the primitive (single) Gaussians, and then the rest can be done using summations over coefficients.
These two orbitals are the most commonly used type of orbitals in Quantum Chemistry but they are not the only basis that can be used. Plane waves are commonly applied to solid state physics because they describe interactions for all the parts in the lattice but provide no information about the positions of the atoms. The basis required for a plane wave set must be very large but the integrals required for these are extremely simple. Plane waves sets are sometimes used as intermediate steps for calculating GTOs and STOs as they integrate very simply.
Hartree-Fock Method
Recalling the full electronic Hamiltonian of an N electron system
� = ℎ + �!! ! 1 1 �!! = 2 �!" !"
! Our electronic wavefunction for this will be a single normalized Slater determinant Ψ = |�! … �! > !! ! ! because of the required antisymmetry of the electrons. We can minimize the energy of Ψ, only for the lowest state of each symmetry state which in this case is the ground state. We can also do the exact HF calculation which does not require a basis.
To begin we introduce the same fixed basis as before and the same expansion for the molecular orbitals,
�! = �!�!" !
We still need the same integrals as before: hμν and Sμν. However, to account for the electron-electron interaction one also requires so-called two-electron integrals. In so-called “physicist’s” notation:
∗ ∗ 1 �� �!! �� = �� �� = ��!��!�!(�!)�!(�!) �!(�!)�!(�!) �!"
If one permutes the last two labels one gets the integral
∗ ∗ ! �� �!! �� = �� �� = ��!��!�!(�!)�!(�!) �!(�!)�!(�!) !!"
Equations can often be described in a compact form by using the so-called antisymmetrized to-electron integrals.
�� ∥ �� = �� �� − �� ��
This combination arises essentially because the use of antisymmetrized product wave functions and Slater determinants. In Hartree-Fock theory the two terms give rise to the so-called Coulomb and exchange terms respectively. The coulomb term is a classical term which gives the energy and potential due to the repulsion of two electrons. The exchange term is a quantum term with no classical interpretation. The minus sign arises because of the antisymmetry of electrons .
The Density Matrix and the Fock Matrix
Here we will summarize the basic ingredients of Hartree Fock theory, and illustrate the connection to the one-electron problem discussed above. We begin by using a basis of orbitals �! = ! �!�!"then ∗ occupying them. The summation over all orbitals gives us the density matrix: �!" = ! �!"�!". From this we obtain the Fock matrix �!" = ℎ!" + !,! ��| �� �!". The Fock Matrix is a one-body effective Hamiltonian and arises as an approximation to the true Hamiltonian of the system and can provide good approximations to the orbital energy of the system.
If we know the Fock matrix we can diagonalize it and find the occupied orbitals:
�ℂ = �ℂ�
ℂ†�ℂ = 1
From this we can find the Density matrix which also gives us the Fock Matrix which can then use to find a better Fock matrix which can give a better Density matrix. This process is repeated until the Density matrix converges at which point the trace of the Fock matrix + our one-electron Hamiltonian give the energy. 1 � = �� � + � � !" 2