Introduction
In mathematics, the matrix representation \(\mathbf{A}\) of a linear operator \(\hat{A}\) in a (not necessarily orthonormal) basis \(\{|\mu\rangle\}\) is defined through the equation
\[ \hat{A}|\mu\rangle = \sum_\nu |\nu\rangle A_{\nu\mu}.\]
In our case, we are interested in the non-orthonormal atomic orbital (AO) basis employed in the majority of molecular quantum chemistry programs.
Furthermore, if \(|\psi_i\rangle\) is a molecular orbital (MO), then the column vector \(\mathbf{C}_i\) whose components are defined through the equation
\[ |\psi_i\rangle = \sum_\mu |\mu\rangle C_{\mu i} \]
is the basis representation of \(|\psi_i\rangle\). \(\mathbf{C}_i\) is the \(i\)th column of the full MO coefficient matrix \(\mathbf{C}\). There is an isomorphism between the original orbitals and operators and their matrix and vector representations. Isomorphism means that equations have the same “structure” in both representations, e.g., if the following eigenvalue equation is fulfilled,
\[ \hat{A} |\psi_i\rangle = a|\psi_i\rangle, \]
then we also have
\[ \mathbf{A} \mathbf{C}_i = a\mathbf{C}_i. \]
Quantum-chemical formulas do not fulfill this expectation: when expressing them in the AO basis, the overlap matrix \(\mathbf{S}\) usually appears in one form or another. The reason is that in quantum chemistry, we define the matrix representation of an operator differently than in mathematics, namely as
\[A^\text{AO}_{\nu\mu} = \langle\nu|\hat{A}|\mu\rangle.\]
In this post, I will show how one can understand this difference by introducing the concept of dual basis functions, which allow one to transform equations to the AO basis without too much thinking. This point of view is probably not original, but I have not seen it in textbooks or journal articles before. I hope others will find it useful as well.
The dual basis
The AO basis is not orthonormal. That is, the overlap matrix defined through
\[ S_{\mu\nu} = \langle \mu|\nu\rangle \]
is not equal to the identity matrix. We can define the dual basis functions \(\{|\tilde{\mu}\rangle\}\) by requiring them to be biorthogonal to the original basis functions, i.e.,
\[ \langle \tilde{\mu}|\nu\rangle = \delta_{\mu\nu}. \]
This leads to the explicit equation
\[ |\tilde{\mu}\rangle = \sum_\nu |\nu\rangle S_{\nu\mu}^{-1}. \]
In the following figure, the dual basis concept is illustrated for vectors in the plane. The dual basis is well known from crystallography, where the reciprocal lattice vectors are dual to the real-space lattice vectors.
Using the original and dual basis functions, one can express the resolution of the identity in four different ways,
\[ \hat{1} = \sum_\mu |\tilde{\mu}\rangle\langle \mu| = \sum_\mu |\mu\rangle\langle \tilde{\mu}| = \sum_{\mu\nu} |\mu\rangle S_{\mu\nu}^{-1} \langle \nu| = \sum_{\mu\nu} |\tilde{\mu}\rangle S_{\mu\nu} \langle \tilde{\nu}|. \]
Furthermore, we can obtain the “mathematical” matrix and vector representations of operators and orbitals by projecting with dual basis functions from the left on the defining equations,
\[ \langle \tilde{\nu}|\psi_i\rangle = \sum_\mu \underbrace{\langle\tilde{\nu}|\mu\rangle}_{=\delta_{\nu\mu}} C_{\mu i} = C_{\nu i}, \]
\[ \langle \tilde{\kappa}|\hat{A}|\mu\rangle = \sum_\nu \underbrace{\langle \tilde{\kappa}|\nu\rangle}_{=\delta_{\kappa\nu}} A_{\nu\mu} = A_{\kappa\mu}. \]
Compare the last equation with the usual quantum chemistry definition of \(\mathbf{A}^\text{AO}\) above. In the mathematical definition of the basis representation, we project from the left with a dual basis function instead of a normal basis function.
Examples
Density matrix
We first consider equations involving the spatial one-particle reduced density operator and its basis representation. If \(\hat{A}\) is a spatial one-electron operator, e.g., a component of the dipole moment, then its expectation value can be written as
\[ \langle \hat{A}\rangle = \text{tr}(\hat{D} \hat{A}) = \text{tr}(\mathbf{D}^\text{AO} \mathbf{A}^\text{AO}). \]
This is an example where the equation for the original operators and their matrix representations are the same even when using the quantum chemistry definitions. Let us investigate this further by starting with the operator equation and evaluating the trace in the AO basis, for which we must employ the mathematical definition of the matrix representation,
\[ \text{tr}(\hat{D}\hat{A}) = \sum_\mu \langle \tilde{\mu} | \hat{D} \hat{A}|\mu\rangle = \sum_{\mu\nu} \langle \tilde{\mu} | \hat{D} | \tilde{\nu}\rangle \underbrace{\langle \nu|\hat{A}|\mu\rangle}_{=A^\text{AO}_{\nu\mu}}. \]
In the second step, we inserted one of the representations of the identity operator introduced above. This shows that what we call the AO density matrix in quantum chemistry is different from other operators in an important aspect: it is a matrix element of the density operator between dual basis functions rather than the original basis functions:
\[ D^\text{AO}_{\mu\nu} = \langle \tilde{\mu} | \hat{D} | \tilde{\nu}\rangle. \]
I found this quite surprising when I first realized it.
Let us do some sanity checks. The restricted Hartree–Fock (RHF) spatial density operator can be expressed via the following spectral decomposition:
\[ \hat{D} = 2\sum_{i\in\text{occ.}}|\psi_i\rangle\langle \psi_i|. \]
Sandwiching this equation between dual basis functions from the left and the right gives
\[ D^\text{AO}_{\mu\nu} = \langle \tilde{\mu}|\hat{D}|\tilde{\nu}\rangle = 2\sum_{i\in\text{occ.}} \langle \tilde{\mu}|\psi_i\rangle \langle \psi_i|\tilde{\nu}\rangle = 2\sum_{i\in\text{occ.}} C_{\mu i} C^\ast_{\nu i}. \]
This is the well-known expression for the RHF AO density matrix in terms of MO coefficients.
As another example, consider the pseudo-idempotency of the RHF spatial density operator,
\[ \hat{D}^2 = 2\hat{D}. \]
To pass to the AO basis, we project with a dual basis function from the left and from the right and furthermore insert a resolution of the identity involving only dual basis functions to obtain
\[ \sum_{\lambda\kappa} \langle \tilde{\mu}|\hat{D}|\tilde{\lambda}\rangle S_{\lambda\kappa} \langle \tilde{\kappa} |\hat{D}|\tilde{\nu}\rangle = 2\langle \tilde{\mu}|\hat{D}|\tilde{\nu}\rangle. \]
This can also be written as
\[ \mathbf{D}\mathbf{S}\mathbf{D} = 2\mathbf{D}. \]
Again, this is a well-known equation in RHF theory.
Roothan–Hall equations
As a second example, consider the RHF self-consistent field equations
\[ \hat{F} |\psi_i\rangle = \epsilon_i |\psi_i\rangle. \]
The AO Fock matrix is defined as a matrix element between normal basis functions, so we first project from the left with a normal basis function to obtain
\[ \langle \mu|\hat{F} |\psi_i\rangle = \epsilon_i \langle \mu|\psi_i\rangle. \]
Next, we can insert a resolution of the identity on both sides of the equation to obtain
\[ \sum_\nu \langle \mu|\hat{F}|\nu\rangle \langle \tilde{\nu} |\psi_i\rangle = \epsilon_i \sum_\nu \langle \mu|\nu\rangle \langle \tilde{\nu}|\psi_i\rangle, \]
which can also be written as
\[ \mathbf{F}^\text{AO} \mathbf{C}_i = \epsilon_i \mathbf{S} \mathbf{C}_i. \]
These are the Roothan–Hall equations of RHF theory.
Summary
The main points of this post are:
- In quantum chemistry, the matrix representation of an operator is defined differently than in standard mathematics.
- As a result, quantum-chemical equations have a different form in the non-orthonormal AO basis, usually involving the overlap matrix \(\mathbf{S}\).
- The concept of dual basis functions makes it easy to understand this difference and to derive equations in the AO basis.
I hope you find the concept of dual basis functions as useful as I do.