Brian Hoffman's Quantum Chemistry Notes

Configuration Interaction and
Coupled Cluster Methods

CI Energy Expression and Density Matrices

The theoretical basis of the CI method has been introduced from the distinct viewpoints of matrix mechanics and functional minimization, but so far many of the details of the CI method have not been addressed. In order to obtain a better understanding of the CI method let us derive an expression for the CI energy in terms of the one- and two-electron integrals. According to the Average Value Theorem, the expectation value of the energy is

$\begin{displaymath}E = \langle \Psi \vert \hat{H} \vert \Psi \rangle. \end{displaymath}$

(26)

Substituting in the linear expansions of Slater determinants for $\Psi$ and taking advantage of the shorthand notation of section 1.10 yields

$\begin{displaymath}E = \langle \sum_i C_I \Phi_I \vert \hat{h} \vert \sum_J C_J \Phi_J \rangle = \sum_{IJ} C_I C_J H_{IJ}. \end{displaymath}$

(27)

Taking the next step and expanding the Hamiltonian matrix elements in terms of one- and two-electron spatial integrals by using Slater's rules and doing any necessary spin integration yields

$\begin{displaymath}E = \sum_I C_I C_J \left[ \sum_{ij}^{MOs} \gamma_{ij}^{IJ} h_... ...{2} \sum_{ijkl}^{MOs} \Gamma_{ijkl}^{IJ} (ij \mid kl) \right]. \end{displaymath}$

(28)

The quantities $\gamma_{ij}^{IJ}$ and $\Gamma_{ijkl}^{IJ}$ are the one- and two-electron coupling constants, respectively, and are simple quantities such as $\frac{1}{2}$ , 1, or 2 which are determined from Slater's rules. In many texts the definition of $\Gamma_{ijkl}^{IJ}$ is adjusted to incorporate the factor of $\frac{1}{2}$ outside the last summation. The coupling constants may be combined with the CI coefficients to produce the one-particle density matrix (OPDM),

$\begin{displaymath}\gamma_{ij} = \sum_{IJ}^{CI} C_I C_J \gamma_{ij}^{IJ}, \end{displaymath}$

(29)

and the two-particle density matrix (TPDM),

$\begin{displaymath}\gamma_{ijkl} = \sum_{IJ}^{CI} C_I C_J \Gamma_{ijkl}^{IJ}. \end{displaymath}$

(30)

In terms of the one- and two-particle density matrices, the energy expression may be written as

$\begin{displaymath}E = \sum_{ij}^{MOs} \gamma_{ij} h_{ij} +\frac{1}{2} \sum_{ijkl}^{MOs} \Gamma_{ijkl} (ij \mid kl). \end{displaymath}$

(31)

Much like its SCF counterpart, the CI OPDM is instrumental in determining a variety of one-electron properties such as the dipole moment. The TPDM has minimal importance for the determination of properties other than the energy. In order to understand the exact nature of the OPDM and TPDM, it is helpful to reformulate the CI problem in the notation of second quantization. In the second quantization formalism, the wave function is written as a product of creation and annihilation operators which act upon a vacuum reference state. Creation operators, denoted $a^{\dagger}_i$ , create an electron in the i^thspin orbital; whereas, annihilation operators, denoted a_i, destroy an electron in the i^th spin orbital. In second quantization, the properties which a many electron wave function must satisfy are converted to a set of properties which must be obeyed by these operators. In order to be physically realistic, a creation operator which acts upon a wave function in which the spin orbital is already occupied must yield a zero result. Moreover, an annihilation operator which destroys an electron which does not exist must produce a zero result. Thus, any annihilation operator acting on the vacuum state must give a zero result. From these basic properties and the Antisymmetry Principle, it can be shown that creation and annihilation operators must satisfy the following anti-commutation relations:

$\displaystyle \{a_j,\; a_i \}$	= a_j a_i+ a_i a_j	= 0	(32)
$\displaystyle \{ {a_j}^{\dagger}, \; {a_i}^{\dagger} \}$	$\textstyle = {a_j}^{\dagger} {a_i}^{\dagger} + {a_i}^{\dagger} {a_j}^{\dagger}$	= 0	(33)
$\displaystyle \{a_i, \; {a_j}^{\dagger} \}$	$\textstyle = a_i {a_j}^{\dagger} + {a_j}^{\dagger} a_i$	$\displaystyle = \delta_{ij} .$	(34)

It is possible to write many of the operators encountered in this dissertation in a second-quantized form. Most notably, the electronic Hamiltonian may be rewritten as

$\begin{displaymath}\hat{H} = \sum_{pq}^{2n} {a_p}^{\dagger} a_q [p\mid \hat{h} \... ...^{2n} {a_p}^{\dagger} {a_r}^{\dagger} a_s a_q [pq \mid rs ] . \end{displaymath}$

(35)

It is interesting to note that in this form, the electronic Hamiltonian is completely independent of the number of electrons. Furthermore, if the spatial components of the $\alpha$ and $\beta$ spin orbitals are identical, as in the restricted formalism, then the second-quantized Hamiltonian may be rewritten as follows in terms of the shift or replacement operator $\hat{E}_{ij}$ ,

$\begin{displaymath}\hat{E}_{ij} = {a_{i \alpha}}^\dagger a_{j \alpha} + {a_{i \beta}} a_{j \beta}, \end{displaymath}$

(36)

which replaces the j^th spatial orbital with the i^th spatial orbital:

$\begin{displaymath}\hat{H} = \sum_{pq}^n h_{pq} \hat{E}_pq + \frac{1}{2} \sum_{p... ...hat{E}_{pq} \hat{E}_{rs} - \delta_{qr} \hat{E}_{ps} \right). \end{displaymath}$

(37)

Having rewritten the electronic Hamiltonian in a convenient second-quantized form, it should become possible to derive clearer second-quantized definitions of the one- and two-particle coupling constants by inserting the rewritten Hamiltonian into the average value expression for the energy:

E	=	$\displaystyle \langle \Psi \vert \hat{H} \vert \Psi \rangle$	(38)
	=	$\displaystyle \langle \sum_I C_I \Phi_I \vert \hat{H} \vert \sum_J C_J \Phi_J \rangle$	(39)
	=	$\displaystyle \sum_{IJ} C_I C_J \langle \Phi_I \vert \sum_{pq}^n h_{pq} \hat{E}_pq$
		$\displaystyle + \frac{1}{2} \sum_{pqrs}^n (pq \mid rs) \left( \hat{E}_{pq} \hat{E}_{rs} - \delta_{qr} \hat{E}_{ps} \right) \vert \Phi_J \rangle$	(40)
	=	$\displaystyle \sum C_I C_J \sum_{pq}^n \langle \Phi_I \vert \hat{E}_{pq} \vert \Phi_J \rangle h_{pq}$
		$\displaystyle + \frac{1}{2} \sum_{pqrs}^n \langle \Phi_I \vert \left( \hat{E}_{... ...t{E}_{rs} - \delta_{qr} \hat{E}_{ps} \right) \vert \Phi_J \rangle (pq \mid rs).$	(41)

Indeed, comparison of the energy equations (2.31) and (2.41) reveals that the one- and two-particle coupling constants are equal to

$\displaystyle \gamma_{ij}^{IJ}$	=	$\displaystyle \langle \Phi_I \vert \hat{E}_{pq} \vert \Phi_J \rangle$	(42)
$\displaystyle {\rm and } \; \Gamma{ijkl}^{IJ}$	=	$\displaystyle \langle \Phi_I \vert \left( \hat{E}_{pq} \hat{E}_{rs} - \delta_{qr} \hat{E}_{ps} \right) \vert \Phi_J \rangle .$	(43)

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