HATREE
, An Introduction to Hartree-Fock Molecular Orbital
Theory
C. David Sherrill
School of Chemistry and Biochemistry
Georgia Institute of Technology
June 2000
1 Introduction
Hartree-Fock theory is fundamental to much of electronic structure theory. It is the basis of
molecular orbital (MO) theory, which posits that each electron’s motion can be described by a
single-particle function (orbital) which does not depend explicitly on the instantaneous motions
of the other electrons. Many of you have probably learned about (and maybe even solved prob-
lems with) Hückel MO theory, which takes Hartree-Fock MO theory as an implicit foundation and
throws away most of the terms to make it tractable for simple calculations. The ubiquity of orbital
concepts in chemistry is a testimony to the predictive power and intuitive appeal of Hartree-Fock
MO theory. However, it is important to remember that these orbitals are mathematical constructs
which only approximate reality. Only for the hydrogen atom (or other one-electron systems, like
He+ ) are orbitals exact eigenfunctions of the full electronic Hamiltonian. As long as we are content
to consider molecules near their equilibrium geometry, Hartree-Fock theory often provides a good
starting point for more elaborate theoretical methods which are better approximations to the elec-
tronic Schrödinger equation (e.g., many-body perturbation theory, single-reference configuration
interaction). So...how do we calculate molecular orbitals using Hartree-Fock theory? That is the
subject of these notes; we will explain Hartree-Fock theory at an introductory level.
2 What Problem Are We Solving?
It is always important to remember the context of a theory. Hartree-Fock theory was developed
to solve the electronic Schrödinger equation that results from the time-independent Schrödinger
equation after invoking the Born-Oppenheimer approximation. In atomic units, and with r de-
noting electronic and R denoting nuclear degrees of freedom, the electronic Schrödinger equation
1
,is
−
1 X 2 X ZA X ZA ZB X 1
∇i − + + Ψ(r; R) = Eel Ψ(r; R), (1)
2 i A,i rAi A>B RAB i>j rij
or, in our previous more compact notation,
h i
T̂e (r) + V̂eN (r; R) + V̂N N (R) + V̂ee (r) Ψ(r; R) = Eel Ψ(r; R). (2)
Recall from the Born-Oppenheimer approximation that Eel (plus or minus V̂N N (R), which we in-
clude here) will give us the potential energy experienced by the nuclei. In other words, E el (R) gives
the potential energy surface (from which we can get, for example, the equilibrium geometry and
vibrational frequencies). That’s one good reason why we want to solve the electronic Schrödinger
equation. The other is that the electronic wavefunction Ψ(r; R) contains lots of useful information
about molecular properties such as dipole (and multipole) moments, polarizability, etc.
3 Motivation and the Hartree Product
The basic idea of Hartree-Fock theory is as follows. We know how to solve the electronic problem
for the simplest atom, hydrogen, which has only one electron. We imagine that perhaps if we
added another electron to hydrogen, to obtain H− , then maybe it might be reasonable to start
off pretending that the electrons don’t interact with each other (i.e., that V̂ee = 0). If that was
true, then the Hamiltonian would be separable, and the total electronic wavefunction Ψ(r 1 , r2 )
describing the motions of the two electrons would just be the product of two hydrogen atom
wavefunctions (orbitals), ΨH (r1 )ΨH (r2 ) (you should be able to prove this easily).
Obviously, pretending that the electrons ignore each other is a pretty serious approximation!
Nevertheless, we have to start somewhere, and it seems plausible that it might be useful to start
with a wavefunction of the general form
ΨHP (r1 , r2 , · · · , rN ) = φ1 (r1 )φ2 (r2 ) · · · φN (rN ), (3)
which is known as a Hartree Product.
While this functional form is fairly convenient, it has at least one major shortcoming: it fails
to satisfy the antisymmetry principle, which states that a wavefunction describing fermions should
be antisymmetric with respect to the interchange of any set of space-spin coordinates. By space-
spin coordinates, we mean that fermions have not only three spatial degrees of freedom, but also
an intrinsic spin coordinate, which we will call α or β. We call a generic (either α or β) spin
coordinate ω, and the set of space-spin coordinates x = {r, ω}. We will also change our notation
for orbitals from φ(r), a spatial orbital, to χ(x), a spin orbital. Except in strange cases such as
2
, the so-called General Hartree Fock or Z-Averaged Perturbation Theory, usually the spin orbital is
just the product of a spatial orbital and either the α or β spin function, i.e., χ(x) = φ(r)α. [Note:
some textbooks write the spin function formally as a function of ω, i.e., α(ω)].
More properly, then, with the full set of coordinates, the Hartree Product becomes
ΨHP (x1 , x2 , · · · , xN ) = χ1 (x1 )χ2 (x2 ) · · · χN (xN ). (4)
This wavefunction does not satisfy the antisymmetry principle! To see why, consider the case for
only two electrons:
ΨHP (x1 , x2 ) = χ1 (x1 )χ2 (x2 ). (5)
What happens when we swap the coordinates of electron 1 with those of electron 2?
ΨHP (x2 , x1 ) = χ1 (x2 )χ2 (x1 ). (6)
The only way that we get the negative of the original wavefunction is if
χ1 (x2 )χ2 (x1 ) = −χ1 (x1 )χ2 (x2 ), (7)
which will not be true in general! So we can see the Hartree Product is actually very far from
having the properties we require.
4 Slater Determinants
For our two electron problem, we can satisfy the antisymmetry principle by a wavefunction like:
1
Ψ(x1 , x2 ) = √ [χ1 (x1 )χ2 (x2 ) − χ1 (x2 )χ2 (x1 )] . (8)
2
This is very nice because it satisfies the antisymmetry requirement for any choice of orbitals χ 1 (x)
and χ2 (x).
What if we have more than two electrons? We can generalize the above solution to N electrons
by using determinants. In the two electron case, we can rewrite the above functional form as
¯ ¯
1 ¯ χ (x ) χ2 (x1 ) ¯
Ψ(x1 , x2 ) = √ ¯¯¯ 1 1 ¯
(9)
2 χ1 (x2 ) χ2 (x2 )
¯
¯
Note a nice feature of this; if we try to put two electrons in the same orbital at the same time
(i.e., set χ1 = χ2 ), then Ψ(x1 , x2 ) = 0. This is just a more sophisticated statement of the Pauli
exclusion principle, which is a consequence of the antisymmetry principle!
3
, An Introduction to Hartree-Fock Molecular Orbital
Theory
C. David Sherrill
School of Chemistry and Biochemistry
Georgia Institute of Technology
June 2000
1 Introduction
Hartree-Fock theory is fundamental to much of electronic structure theory. It is the basis of
molecular orbital (MO) theory, which posits that each electron’s motion can be described by a
single-particle function (orbital) which does not depend explicitly on the instantaneous motions
of the other electrons. Many of you have probably learned about (and maybe even solved prob-
lems with) Hückel MO theory, which takes Hartree-Fock MO theory as an implicit foundation and
throws away most of the terms to make it tractable for simple calculations. The ubiquity of orbital
concepts in chemistry is a testimony to the predictive power and intuitive appeal of Hartree-Fock
MO theory. However, it is important to remember that these orbitals are mathematical constructs
which only approximate reality. Only for the hydrogen atom (or other one-electron systems, like
He+ ) are orbitals exact eigenfunctions of the full electronic Hamiltonian. As long as we are content
to consider molecules near their equilibrium geometry, Hartree-Fock theory often provides a good
starting point for more elaborate theoretical methods which are better approximations to the elec-
tronic Schrödinger equation (e.g., many-body perturbation theory, single-reference configuration
interaction). So...how do we calculate molecular orbitals using Hartree-Fock theory? That is the
subject of these notes; we will explain Hartree-Fock theory at an introductory level.
2 What Problem Are We Solving?
It is always important to remember the context of a theory. Hartree-Fock theory was developed
to solve the electronic Schrödinger equation that results from the time-independent Schrödinger
equation after invoking the Born-Oppenheimer approximation. In atomic units, and with r de-
noting electronic and R denoting nuclear degrees of freedom, the electronic Schrödinger equation
1
,is
−
1 X 2 X ZA X ZA ZB X 1
∇i − + + Ψ(r; R) = Eel Ψ(r; R), (1)
2 i A,i rAi A>B RAB i>j rij
or, in our previous more compact notation,
h i
T̂e (r) + V̂eN (r; R) + V̂N N (R) + V̂ee (r) Ψ(r; R) = Eel Ψ(r; R). (2)
Recall from the Born-Oppenheimer approximation that Eel (plus or minus V̂N N (R), which we in-
clude here) will give us the potential energy experienced by the nuclei. In other words, E el (R) gives
the potential energy surface (from which we can get, for example, the equilibrium geometry and
vibrational frequencies). That’s one good reason why we want to solve the electronic Schrödinger
equation. The other is that the electronic wavefunction Ψ(r; R) contains lots of useful information
about molecular properties such as dipole (and multipole) moments, polarizability, etc.
3 Motivation and the Hartree Product
The basic idea of Hartree-Fock theory is as follows. We know how to solve the electronic problem
for the simplest atom, hydrogen, which has only one electron. We imagine that perhaps if we
added another electron to hydrogen, to obtain H− , then maybe it might be reasonable to start
off pretending that the electrons don’t interact with each other (i.e., that V̂ee = 0). If that was
true, then the Hamiltonian would be separable, and the total electronic wavefunction Ψ(r 1 , r2 )
describing the motions of the two electrons would just be the product of two hydrogen atom
wavefunctions (orbitals), ΨH (r1 )ΨH (r2 ) (you should be able to prove this easily).
Obviously, pretending that the electrons ignore each other is a pretty serious approximation!
Nevertheless, we have to start somewhere, and it seems plausible that it might be useful to start
with a wavefunction of the general form
ΨHP (r1 , r2 , · · · , rN ) = φ1 (r1 )φ2 (r2 ) · · · φN (rN ), (3)
which is known as a Hartree Product.
While this functional form is fairly convenient, it has at least one major shortcoming: it fails
to satisfy the antisymmetry principle, which states that a wavefunction describing fermions should
be antisymmetric with respect to the interchange of any set of space-spin coordinates. By space-
spin coordinates, we mean that fermions have not only three spatial degrees of freedom, but also
an intrinsic spin coordinate, which we will call α or β. We call a generic (either α or β) spin
coordinate ω, and the set of space-spin coordinates x = {r, ω}. We will also change our notation
for orbitals from φ(r), a spatial orbital, to χ(x), a spin orbital. Except in strange cases such as
2
, the so-called General Hartree Fock or Z-Averaged Perturbation Theory, usually the spin orbital is
just the product of a spatial orbital and either the α or β spin function, i.e., χ(x) = φ(r)α. [Note:
some textbooks write the spin function formally as a function of ω, i.e., α(ω)].
More properly, then, with the full set of coordinates, the Hartree Product becomes
ΨHP (x1 , x2 , · · · , xN ) = χ1 (x1 )χ2 (x2 ) · · · χN (xN ). (4)
This wavefunction does not satisfy the antisymmetry principle! To see why, consider the case for
only two electrons:
ΨHP (x1 , x2 ) = χ1 (x1 )χ2 (x2 ). (5)
What happens when we swap the coordinates of electron 1 with those of electron 2?
ΨHP (x2 , x1 ) = χ1 (x2 )χ2 (x1 ). (6)
The only way that we get the negative of the original wavefunction is if
χ1 (x2 )χ2 (x1 ) = −χ1 (x1 )χ2 (x2 ), (7)
which will not be true in general! So we can see the Hartree Product is actually very far from
having the properties we require.
4 Slater Determinants
For our two electron problem, we can satisfy the antisymmetry principle by a wavefunction like:
1
Ψ(x1 , x2 ) = √ [χ1 (x1 )χ2 (x2 ) − χ1 (x2 )χ2 (x1 )] . (8)
2
This is very nice because it satisfies the antisymmetry requirement for any choice of orbitals χ 1 (x)
and χ2 (x).
What if we have more than two electrons? We can generalize the above solution to N electrons
by using determinants. In the two electron case, we can rewrite the above functional form as
¯ ¯
1 ¯ χ (x ) χ2 (x1 ) ¯
Ψ(x1 , x2 ) = √ ¯¯¯ 1 1 ¯
(9)
2 χ1 (x2 ) χ2 (x2 )
¯
¯
Note a nice feature of this; if we try to put two electrons in the same orbital at the same time
(i.e., set χ1 = χ2 ), then Ψ(x1 , x2 ) = 0. This is just a more sophisticated statement of the Pauli
exclusion principle, which is a consequence of the antisymmetry principle!
3
