The antisymmetrizer

Photo Credit: Rene Magritte - Decalcomania

The antisymmetrizer

In this post the antisymmetrizer operator is officially defined.

Those who read the determinant wavefunctions post will already be familiar with this operator.

I discuss it again here in order to describe more of it’s properties, which are important for evaluating many-electron integrals. Such integrals form the bread-and-butter of quantum chemistry — so knowing about the antisymmetrizer is important.

More important for me is that the “antisymmetric” nature of electrons is one of the puzzling and amazing things about quantum mechanics itself. It’s just so cool and weird!

The antisymmetrizer

The antisymmetrizer is so-called because when it is applied to an arbitrary function of labelled arguments, that function is changed into a new function which changes sign whenever those arguments are swapped.

The antisymmetrizer operator is defined by

Notes:

  • The sum is over all permutations \( u \) of a set of integers \( { 1, \ldots, N } \).

  • \( P_u \) is an operator which permutes those integer labels in any expression which follows. For example, \( P_u \) might be the elementary permutation \( (1 2) \) which swaps labels 1 and 2, although there may be more complicated permutations such as the cycle \( (1 2 3) \). See wikpedia here for more information.

  • The number of terms in the summation is \( N! \).

  • \( \epsilon_u \) is a number which is either +1 or -1 called the signature, sign or paity of the permutation \( P_u \). It is +1 (or -1) if \( P_u \) can be decomposed into a sequence of even (rrespectively, odd) elementary permutations or swaps . It is not at all obvious that any permutation may be decomposed as either even or odd. If you are interested, several proofs of this fact are given in wikipedia. The quantity \( \epsilon_u \) is also called the Levi-Civita symbol which may be familiar to some of you.

Making functions antisymmetric

We have the following theorem:

The function \( F(\B{x}_1 \ldots \B{x}_n) = \mathscr{A}\, f(\B{x}_1 \ldots \B{x}_n) \) is antisymmetric, i.e.

Proof:

Remarks: The only tricky part is the penultimate line. There, we used two facts. First, that the product of two permutations is another permutation \( P_v\). This should be obvious, but actually follows from the fact that permutations form a mathematical group. Second, since \( P_v \) has one extra swap compared to \( P_u \) its signature \( \epsilon_v = -\epsilon_u \).

We have shown that the antisymmetrizer \( \mathscr{A} \) makes any function which it is applied to antisymmetric.

The antisymmetrizer makes determinant wavefunctions

If you read the previous post you will see that the antisymmetrizer makes determinant wavefunctions:

The antisymmetrizer is Hermitian

The antisymmetrizer is Hermitian if \( \mathscr{A}^\dagger = \mathscr{A} \), which means that for any functions \( \Phi_I \) and \( \Phi_J \) that these operators are allowed to “act” on we must have

Proof:

Remark: In an \( M \)-dimensional finite dimensional space \( V = \mbox{span}\, (\Phi_I ; I=1,\ldots,M) \) the adjoint \( \mathscr{A}^\dagger \) is also called the “hermitian conjugate”. In quantum chemistry we usually deal with finite dimensional spaces, so the terms “self-adjoint” and “hermitian” are equivalent. To be honest I am no sure of the relevance of infinite dimensional spaces in any practical calculation.

Antisymmetrizers are (nearly) idempotent

The following theorem holds:

In other words, antisymmetrizing a function which is already antisymmetric doesn’t change it from being antisymmetric (except for a factor).

Note: An operators which obeys \( P^2 = P \) is called an idempotent or projection operator. In our case \( (1/\sqrt{N!}) \mathscr{A} \) would be idempotent.

Proof: