4.6 Non-orthogonal orbitals

We conclude this chapter with a discussion about the representation of the density-matrix using non-orthogonal orbitals. We consider a set of non-orthogonal functions $\{ \phi_{\alpha}({\bf r}) \}$ which we denote $\{ \vert \phi_{\alpha} \rangle \}$, and introduce their dual functions defined by

$$\vert \phi^{\alpha} \rangle = \vert \phi_{\beta} \rangle S_{\beta \alpha}^{-1}$$ (4.69)

in which the summation convention is assumed and the matrix $S^{-1}$ is the inverse of the overlap matrix $S$ defined by

$$S_{\alpha \beta} = \langle \phi_{\alpha} \vert \phi_{\beta} ... ...athrm d}{\bf r}~\phi_{\alpha}({\bf r}) \phi_{\beta}({\bf r}) .$$ (4.70)

We have assumed from now on that we are only calculating wave-functions at the $\Gamma$-point and so can assume that everything is real. By construction, the dual states obey

$$\langle \phi^{\alpha} \vert \phi_{\beta} \rangle = \langle \phi_{\alpha} \vert \phi^{\beta} \rangle = \delta_{\alpha}^{\beta}$$ (4.71)

and the completeness relation is expressed as

$$\vert \phi^{\alpha} \rangle \langle \phi_{\alpha} \vert = \v... ...\rangle S_{\alpha \beta}^{-1} \langle \phi_{\beta} \vert = 1 .$$ (4.72)

In general we will represent the density-matrix in the separable form

$$\rho({\bf r},{\bf r'}) = \phi_{\alpha}({\bf r}) K^{\alpha \beta} \phi_{\beta} ({\bf r'})$$ (4.73)

and note that the density-kernel $K^{\alpha \beta} \not= \langle \phi_{\alpha} \vert {\hat \rho} \vert \phi_{\beta} \rangle$ because of the non-orthogonality.

We can construct an orthonormalised set of orbitals $\{ \vert\varphi_{\alpha}\rangle \}$ defined as linear combinations of the $\{ \vert \phi_{\alpha} \rangle \}$ by the Löwdin transformation:

$$\vert \varphi_{\alpha} \rangle = \vert \phi_{\beta} \rangle S_{\beta \alpha}^{-{1 \over 2}}$$ (4.74)

such that

$$\langle \varphi_{\alpha} \vert \varphi_{\beta} \rangle = S_{... ...lta} S_{\delta \beta}^{-{1 \over 2}} = \delta_{\alpha \beta} .$$ (4.75)

At the ground-state, these orthonormal orbitals $\{ \vert\varphi_{\alpha}\rangle \}$ will be a unitary transformation of the Kohn-Sham orbitals $\{ \vert \psi_i \rangle \}$, so that the density-kernel ${\tilde K}$ defined as the density-matrix in the representation of the orthonormalised orbitals $\{ \vert\varphi_{\alpha}\rangle \}$,

$${\tilde K}_{\alpha \beta} = \langle \varphi_{\alpha} \vert {\hat \rho} \vert \varphi_{\beta} \rangle ,$$ (4.76)

can be diagonalised by a unitary transformation $U$ as described in section 4.5.1 i.e. $f_i = (U^{\dag } {\tilde K} U)_{ii}$ (no summation convention). The following relationship also holds;

$$\vert \psi_i \rangle = \vert \varphi_{\alpha} \rangle U_{\alpha i} ,$$ (4.77)

and the relationship between the non-orthogonal orbitals $\{ \vert \phi_{\alpha} \rangle \}$ and the Kohn-Sham orbitals $\{ \vert \psi_i \rangle \}$ is of the general form

$$\vert \psi_i \rangle = \vert \varphi_{\alpha} \rangle U_{\al... ...over 2}} U_{\alpha i} = \vert \phi_{\beta} \rangle V_{\beta i}$$ (4.78)

where the matrix $V = S^{-{1 \over 2}} U$ and obeys

$$V^{\dag } V = U^{\dag } S^{-1} U ,$$ (4.79)

$$V V^{\dag } = S^{-1} .$$ (4.80)

Using the completeness relation (4.72) we can now express the matrix $K$ in terms of other quantities:

$$\rho({\bf r},{\bf r'}) = \phi_{\alpha}({\bf r}) K^{\alpha \beta} \phi_{\beta}({\bf r'})$$

$$= \langle {\bf r} \vert {\hat \rho} \vert {\bf r'} \rangle = \langl... ...elta} \rangle S_{\delta \beta}^{-1} \langle \phi_{\beta} \vert {\bf r'} \rangle$$

$$= \phi_{\alpha}({\bf r}) S_{\alpha \gamma}^{-1} \langle \phi_{\gamm... ... \rho} \vert \phi_{\delta} \rangle S_{\delta \beta}^{-1} \phi_{\beta}({\bf r'})$$ (4.81)

so that

$$K^{\alpha \beta} = S_{\alpha \gamma}^{-1} \langle \phi_{\gamma} \vert {\hat \rho} \vert \phi_{\delta} \rangle S_{\delta \beta}^{-1} ,$$ (4.82)

$$\langle \phi_{\alpha} \vert {\hat \rho} \vert \phi_{\beta} \rangle = (SKS)_{\alpha \beta} .$$ (4.83)

In fact, the density-kernel $K$ contains the matrix elements of the density-operator in the representation of the dual vectors of the non-orthogonal functions:

$$K^{\alpha \beta} = S_{\alpha \gamma}^{-1} \langle \phi_{\gam... ...e \phi^{\alpha} \vert {\hat \rho} \vert \phi^{\beta} \rangle ,$$ (4.84)

hence the superscript notation.

If we wish to obtain the occupation numbers, we must diagonalise the matrix ${\tilde K}$ which is given by

$${\tilde K}_{\alpha \beta} = \langle \varphi_{\alpha} \vert {\hat \rho} \vert \varphi_{\beta} \rangle$$

$$= \langle \varphi_{\alpha} \vert \phi_{\gamma} \rangle S_{\gamma \d... ...ngle S_{\epsilon \zeta}^{-1} \langle \phi_{\zeta} \vert \varphi_{\beta} \rangle$$

$$= S_{\alpha \gamma}^{1 \over 2} S_{\gamma \delta}^{-1} (SKS)_{\delta \epsilon} S_{\epsilon \zeta}^{-1} S_{\zeta \beta}^{1 \over 2}$$

$$= (S^{1 \over 2} K S^{1 \over 2})_{\alpha \beta} .$$ (4.85)

Thus the eigenvalues of $(S^{1 \over 2} K S^{1 \over 2})$ are the occupation numbers.

At the ground-state, the density-operator and Hamiltonian commute, and thus both the Hamiltonian and the density-matrix can be diagonalised simultaneously. The Hamiltonian is usually represented by its matrix elements in the representation of the non-orthogonal orbitals. Thus

$$H_{\alpha \beta} = \langle \phi_{\alpha} \vert {\hat H} \vert \phi_{\beta} \rangle$$ (4.86)

in contrast to the definition of $K$. In this case, to obtain the eigenvalues of the Hamiltonian $\{ \varepsilon_i \}$ it is necessary to diagonalise the matrix ${\tilde H}$

$${\tilde H}_{\alpha \beta} = \langle \varphi_{\alpha} \vert {\hat H} \vert \varphi_{\beta} \rangle$$

$$= \langle \varphi_{\alpha} \vert \phi_{\gamma} \rangle S_{\gamma \d... ...ngle S_{\epsilon \zeta}^{-1} \langle \phi_{\zeta} \vert \varphi_{\beta} \rangle$$

$$= S_{\alpha \gamma}^{1 \over 2} S_{\gamma \delta}^{-1} H_{\delta \epsilon} S_{\epsilon \zeta}^{-1} S_{\zeta \beta}^{1 \over 2}$$

$$= (S^{-{1 \over 2}} H S^{-{1 \over 2}})_{\alpha \beta}$$ (4.87)

i.e. the eigenvalues of $(S^{-{1 \over 2}} H S^{-{1 \over 2}})$ are those of the Kohn-Sham Hamiltonian.

The advantage of representing the density-operator and Hamiltonian in different ways is that quantities such as the electron number and non-interacting energy can be expressed easily:

$$N = 2 {\rm Tr}({\tilde K}) = 2 {\rm Tr}(KS)$$ (4.88)

$$E_{\mathrm{NI}} = 2 {\rm Tr}({\tilde K}{\tilde H}) = 2 {\rm Tr}(KH)$$ (4.89)

since the factors of $S^{-{1 \over 2}}$ and $S^{1 \over 2}$ cancel.

In the language of tensor analysis, the functions $\{ \vert \phi_{\alpha} \rangle \}$ are covariant vectors, and their duals the associated contravariant quantities. The overlap matrix $S_{\alpha \beta}$ plays the rôle of the metric tensor to convert between covariant and contravariant quantities. This is seen by verifying the relationship

$$S^{\alpha \beta} = S^{-1}_{\alpha \beta} .$$ (4.90)

For the orthogonal functions $\{ \vert\varphi_{\alpha}\rangle \}$, the metric tensor is the identity and so there is no distinction between covariant and contravariant quantities.

In a linear-scaling scheme we will not be able to access the eigenvalues directly, since although $H$ and $K$ are sparse, ${\tilde H}$ and ${\tilde K}$ need not be, and in any case, the effort to diagonalise even a sparse matrix is ${\cal O}(N^2)$. However it is important to understand the different origins and rôles of these matrices in order to analyse the equations which result when we attempt to minimise the total energy to find the ground-state.