Many-body perturbation theory
Contents
4. Many-body perturbation theory¶
We assume here that we are only interested in the ground state of the system and expand the exact wave function in term of a series of Slater determinants
where we have assumed that the true ground state is dominated by the solution of the unperturbed problem, that is
The state
The Schroedinger equation is
and multiplying the latter from the left with
and subtracting from this equation
and using the fact that the both operators
which is an exact result. We call this quantity the correlation energy.
This equation forms the starting point for all perturbative derivations. However,
as it stands it represents nothing but a mere formal rewriting of Schroedinger’s equation and is not of much practical use. The exact wave function
Here we have assumed that our model space defined by the operator
and
We can thus rewrite the exact wave function as
Going back to the Schr”odinger equation, we can rewrite it as, adding and a subtracting a term
where
We assume also that the resolvent of
We can rewrite Schroedinger’s equation as
and multiplying from the left with
which is possible since we have defined the operator
These operators commute meaning that
With these definitions we can in turn define the wave function as
This equation is again nothing but a formal rewrite of Schr”odinger’s equation
and does not represent a practical calculational scheme.
It is a non-linear equation in two unknown quantities, the energy
The most common choice is to start with the function which is expected to exhibit the largest overlap with the wave function we are searching after, namely
for the wave function and
which is now a perturbative expansion of the exact energy in terms of the interaction
In our equations for
In Brilluoin-Wigner perturbation theory it is customary to set
1 7
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1 9
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This expression depends however on the exact energy
Actually, the above expression is nothing but a rewrite again of the full Schr”odinger equation.
Defining
2 1
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Inserted in the expression for
In RS perturbation theory we set
2 4
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Recalling that
Inserting this results in the expression for the energy results in
We can now this expression in terms of a perturbative expression in terms
of
We get the following expression for
which is just the contribution to first order in perturbation theory,
which is the contribution to second order.
being the third-order contribution.
4.1. Interpreting the correlation energy and the wave operator¶
In the shell-model lectures we showed that we could rewrite the exact state function for say the ground state, as a linear expansion in terms of all possible Slater determinants. That is, we define the ansatz for the ground state as
where the index
while a
and a general
We use letters
We can then expand our exact state function for the ground state as
where we have introduced the so-called correlation operator
Since the normalization of
resulting in
In a shell-model calculation, the unknown coefficients in
How can we use perturbation theory to determine the same coefficients? Let us study the contributions to second order in the interaction, namely
The intermediate states given by
If we limit the attention to a Hartree-Fock basis, then we have that
If we compare this to the correlation energy obtained from full configuration interaction theory with a Hartree-Fock basis, we found that
where the energy
We see that if we set
we have a perfect agreement between FCI and MBPT. However, FCI includes such
Summing up, we can see that
MBPT introduces order-by-order specific correlations and we make comparisons with exact calculations like FCI
At every order, we can calculate all contributions since they are well-known and either tabulated or calculated on the fly.
MBPT is a non-variational theory and there is no guarantee that higher orders will improve the convergence.
However, since FCI calculations are limited by the size of the Hamiltonian matrices to diagonalize (today’s most efficient codes can attach dimensionalities of ten billion basis states, MBPT can function as an approximative method which gives a straightforward (but tedious) calculation recipe.
MBPT has been widely used to compute effective interactions for the nuclear shell-model.
But there are better methods which sum to infinite order important correlations. Coupled cluster theory is one of these methods.