Bi-orthogonal operators

In order to simplify the calculations in a non-orthogonal spinor basis, we start by defining a set of bi-orthogonal (=reciprocal) orbitals \(\require{physics} \set{\tilde{\chi}_P}\) by the equation \[\begin{equation} \braket{ \tilde{\chi}_P }{ \chi_Q } = \delta_{PQ} % \thinspace , \end{equation}\] which are related to the non-orthogonal set \(\set{\chi_P}\) by means of the transformation \[\begin{equation} \tilde{\chi}_P = \sum_Q^M \chi_Q T_{QP} % \thinspace , \end{equation}\] in which the transformation matrix \(\vb{T}\) is Hermitian and obeys \[\begin{equation} \vb{T} = \vb{S}^{-1} \thinspace , \end{equation}\] such that the transformation to and from the reciprocal set is given by: \[\begin{align} & \tilde{\chi}_P = \sum_Q^M \chi_Q S^{-1}_{QP} \\ & \chi_P = \sum_Q^M \tilde{\chi}_Q S_{QP} % \thinspace . \end{align}\] Therefore, the reciprocal creation operators are expressed as \[\begin{align} & \hat{\tilde{b}}^\dagger_P = \sum_Q^M \hat{b}^\dagger_Q S^{-1}_{QP} \\ & \hat{b}^\dagger_P = \sum_Q^M \hat{\tilde{b}}^\dagger_Q S_{QP} \end{align}\] and their adjoints as \[\begin{align} & \hat{\tilde{b}}_P = \sum_Q^M \hat{b}_Q S^{-1}_{PQ} % \label{eq:bi-orthogonal_operators} \\ & \hat{b}_P = \sum_Q^M \hat{\tilde{b}}_Q S_{PQ} % \thinspace . \end{align}\] By investigating the mixed anticommutator \[\begin{equation} \label{eq:commutator_true_annihilator} \comm{ % \hat{b}_P % }{ % \hat{\tilde{b}}^\dagger_Q % }_+ = % \delta_{PQ} % \thinspace , \end{equation}\] we find that that \(\set{\hat{\tilde{b}}_P}\) are the true annihilators of the creation operators \(\set{\hat{b}_P}\)!

Since this commutator is easy in this framework, we can use results that were derived for orthonormal orbitals based on the anticommutation rule. However, the construction of bra wave functions now becomes harder, since \(\hat{\tilde{b}}_P\) is not the corresponding adjoint of \(\hat{b}^\dagger_P\).

Using the auxiliary orthonormal orbitals (ONOs) \(\set{\phi_P}\), we can write down the Hamiltonian as \[\begin{equation} \hat{\mathcal{H}} = % \sum_{PQ}^M % h^{ \text{ONO} }_{PQ} \hat{E}_{PQ} % + \frac{1}{2} \sum_{PQRS}^M % g^{ \text{ONO} }_{PQRS} \hat{e}_{PQRS} % \thinspace . \end{equation}\] Transforming all operators and integrals in order to obtain an expression in terms of the non-orthogonal creation operators \(\set{\hat{b}^\dagger_P}\) and their adjoints \(\set{\hat{b}_P}\) related to the non-orthogonal orbitals (NOOs) \(\set{\phi_P}\), we get: \[\begin{equation} \begin{split} \hat{\mathcal{H}} = % & \sum_{PQRS}^M % h^{ \text{NOO} }_{RS} % S^{-1}_{PR} S^{-1}_{SQ} % \hat{b}^\dagger_P \hat{b}_Q \\ & + \frac{1}{2} % \sum_{ \substack{PQRS \\ TUVW} }^M % g^{ \text{NOO} }_{TUVW} % S^{-1}_{PT} S^{-1}_{RV} S^{-1}_{WS} S^{-1}_{UQ} % \hat{b}^\dagger_P \hat{b}^\dagger_R % \hat{b}_S \hat{b}_Q % \thinspace . \end{split} \end{equation}\] If we sum over the indices of \(\vb{S}^{-1}\) by introducing the reciprocal operators \(\eqref{eq:bi-orthogonal_operators}\), and the bra-transformed integrals \[\begin{equation} h^{ \text{NOO} }_{\tilde{P} Q} = % \sum_R^M S^{-1}_{PR} h^{\text{NOO}}_{RQ} \end{equation}\] and \[\begin{equation} g^{ \text{NOO} }_{\tilde{P} Q \tilde{R} S} = % \sum_{TU}^M S^{-1}_{PT} S^{-1}_{RU} g^{ \text{NOO} }_{TQUS} % \thinspace , \end{equation}\] we find another equivalent form of the Hamiltonian, but this time using the reciprocal annihilation operators: \[\begin{equation} \hat{\mathcal{H}} = % \sum_{PQ}^M % h^{ \text{NOO} }_{\tilde{P} Q} % \hat{b}^\dagger_P \hat{\tilde{b}}_Q % + \frac{1}{2} \sum_{PQRS}^M % g^{ \text{NOO} }_{\tilde{P} Q \tilde{R} S} % \hat{b}^\dagger_P \hat{b}^\dagger_R % \hat{\tilde{b}}_S \hat{\tilde{b}}_Q % \thinspace . \end{equation}\] This Hamiltonian has the interesting property that it behaves like a representation in an orthonormal basis, due to the appearance of creators and their true annihilators (in the reciprocal space).

If, however, we would instead of introducing the bra-transformed integrals, transform the creation operators to the reciprocal space, we end up with the equivalent Hamiltonian: \[\begin{equation} \hat{\mathcal{H}} = % \sum_{PQ}^M % h^{ \text{NOO} }_{PQ} % \hat{\tilde{b}}^\dagger_P \hat{\tilde{b}}_Q % + \frac{1}{2} \sum_{PQRS}^M % g^{ \text{NOO} }_{PQRS} % \hat{\tilde{b}}^\dagger_P \hat{\tilde{b}}^\dagger_R % \hat{\tilde{b}}_S \hat{\tilde{b}}_Q % \thinspace , \end{equation}\] which again introduces difficult commutation relations (since \(\hat{\tilde{b}}_P\) is not the true annihilator of \(\hat{\tilde{b}}^\dagger_P\)), but the usefulness of this form of the Hamiltonian is that it is ideally suited to define non-orthogonal representations of the 1-DM \[\begin{equation} D^{ \text{NOO} }_{PQ} = % \ev{ % \hat{\tilde{b}}^\dagger_P % \hat{\tilde{b}}_Q % }{\Psi} \end{equation}\] and 2-DM: \[\begin{equation} D^{ \text{NOO} }_{PQRS} = % \ev{ % \hat{\tilde{b}}^\dagger_P \hat{\tilde{b}}^\dagger_R % \hat{\tilde{b}}_S \hat{\tilde{b}}_Q % }{\Psi} % \thinspace . \end{equation}\] We can derive that the \(1\)-DM in the non-orthogonal basis is linked with the one in the orthonormal basis: \[\begin{align} \label{eq:non-orthogonal_1-DM} D^{ \text{NOO} }_{PQ} % &= \sum_{RS}^M % C^*_{PR} C^{\text{T}}_{SQ} D_{RS} \\ &= \qty( % \vb{C}^* \vb{D} \vb{C}^{\text{T}} % )_{PQ} % \thinspace , \end{align}\] which reduces to the well-known formula \[\begin{equation} \vb{D}^{\text{NOO}} = \vb{C} \vb{D} \vb{C}^{\text{T}} \end{equation}\] in the case of real orbitals and a single Slater determinant. We must be cautious, however, because the properties of these density matrices are not the same as those in an orthonormal basis, as \[\begin{equation} \sum_{PQ}^M % D_{PQ}^{\text{AO}} S_{PQ} = N % \thinspace , \end{equation}\] instead of the usual \[\begin{equation} \sum_{P}^M D_{PP} = N % \tag{\ref{eq:trace_1_DM}} % \thinspace . \end{equation}\]