Matrix Elements of ECG Basis Functions

By David K. Zhang


Under Construction!

Consider a system of $N$ particles moving in $d$-dimensional space.

We define the following functions:

(1)
\[\begin{aligned} \Phi_1(\uvx) &\coloneqq \qty[ \prod_{i=1}^N \prod_{k=1}^d x_{ik}^{(p_1)_{ik}} ] \exp( -\frac{1}{2} \uvx^T \! A_1 \uvx ) \\ \Phi_2(\uvx) &\coloneqq \qty[ \prod_{i=1}^N \prod_{k=1}^d x_{ik}^{(p_2)_{ik}} ] \exp( -\frac{1}{2} \uvx^T \! A_2 \uvx ) \end{aligned} \]
(2)
\[\begin{aligned} G_1(\uvx) &\coloneqq \exp( -\frac{1}{2} \uvx^T \! A_1 \uvx + \uvt_1^T \uvx ) \\ G_2(\uvx) &\coloneqq \exp( -\frac{1}{2} \uvx^T \! A_2 \uvx + \uvt_2^T \uvx ) \end{aligned} \]

We make the following definitions:

(3)
\[\begin{aligned} A &\coloneqq A_1 + A_2 \\ \uvt &\coloneqq \uvt_1 + \uvt_2 \end{aligned} \]

The overlap integral of the generating functions is then given by

(4)
\[\braket{G_1}{G_2} = \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \exp(\frac{1}{2} \uvt^T \! A^{-1} \uvt) \]

We now define a $2N \times 2N$ real symmetric matrix $B$ and a $2N$-dimensional vector $\uvu$ whose entries are $d$-dimensional vectors.

(5)
\[B \coloneqq \mqty[ A^{-1} & A^{-1} \\ A^{-1} & A^{-1} ] \qquad \uvu \coloneqq \mqty[ \uvt_1 \\ \uvt_2 ] \]

We now rewrite the overlap integral of the generating functions, as follows:

(6)
\[\begin{aligned} \braket{G_1}{G_2} &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \exp(\frac{1}{2} \uvu^T \! B \uvu) \\ &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \exp(\frac{1}{2} \sum_{i=1}^{2N} \sum_{j=1}^{2N} B_{ij} (\vec{u}_i \cdot \vec{u}_j)) \\ &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \exp(\frac{1}{2} \sum_{i=1}^{2N} B_{ii} (\vec{u}_i \cdot \vec{u}_i) + \sum_{i=1}^{2N} \sum_{j=i+1}^{2N} B_{ij} (\vec{u}_i \cdot \vec{u}_j)) \\ &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \exp(\frac{1}{2} \sum_{i=1}^{2N} \sum_{k=1}^d B_{ii} u_{ik}^2 + \sum_{i<j}^{2N} \sum_{k=1}^d B_{ij} u_{ik} u_{jk}) \\ &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \exp(\sum_{i \le j}^{2N} \sum_{k=1}^d C_{ij} u_{ik} u_{jk}) \end{aligned} \]

Here, we have defined a new $2N \times 2N$ matrix $C$, which is an upper-triangular matrix whose diagonal is one-half times the diagonal of $B$, and whose strictly upper-triangular part is identical to that of $B$.

(7)
\[\begin{aligned} \braket{G_1}{G_2} &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \exp(\sum_{i \le j}^{2N} \sum_{k=1}^d C_{ij} u_{ik} u_{jk}) \\ &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \prod_{i \le j}^{2N} \prod_{k=1}^d \exp(C_{ij} u_{ik} u_{jk}) \\ &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \prod_{i \le j}^{2N} \prod_{k=1}^d \sum_{q_{kij}=0}^\infty \frac{(C_{ij} u_{ik} u_{jk})^{q_{kij}}}{q_{kij}!} \\ &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \sum_{q \in Q} \prod_{k=1}^d \prod_{i \le j}^{2N} \frac{(C_{ij} u_{ik} u_{jk})^{q_{kij}}}{q_{kij}!} \\ \end{aligned} \]

Here we have defined $Q$ as the set of all $d$-tuples of $2N \times 2N$ symmmetric matrices whose entries are nonnegative integers. Thus if $q \in Q$, then $q_{kij}$ denotes the $(i,j)$-entry of the symmetric matrix $q_k$.

(Note that the preceding sum depends only on values of $q_{kij}$ for which $i \le j$; thus, only the upper-triangular part of each matrix $q_k$ is accessed, and we might as well define the lower-triangular part as the transpose of the upper-triangular part.)

Suppose now that we are given an array $q \in Q$ and an integer $1 \le k \le d$. We wish to examine the term

(8)
\[\begin{aligned} \prod_{i \le j}^{2N} \frac{(C_{ij} u_{ik} u_{jk})^{q_{kij}}}{q_{kij}!} &= \prod_{i \le j}^{2N} \frac{C_{ij}^{q_{kij}}}{q_{kij}!} u_{ik}^{q_{kij}} u_{jk}^{q_{kij}} \\ &= \qty[\prod_{i \le j}^{2N} \frac{C_{ij}^{q_{kij}}}{q_{kij}!}] \qty[\prod_{i \le j}^{2N} u_{ik}^{q_{kij}} u_{jk}^{q_{kij}}]. \end{aligned} \]

For each integer $1 \le b \le 2N$, we ask the following question: how many times does $u_{bk}$ appear in the preceding expression?

Thus, the exponent of $u_{bk}$ in the overall expression is

(9)
\[\epsilon_{bk} \coloneqq \sum_{i=1}^{b-1} q_{kib} + 2q_{kbb} + \sum_{j=b+1}^{2N} q_{kbj} \]

which, by using the symmetry of $q_k$, can be rewritten as

(10)
\[\epsilon_{bk} = q_{kbb} + \sum_{i=1}^{2N} q_{kib} \]

This shows that

(11)
\[\prod_{i \le j}^{2N} \frac{(C_{ij} u_{ik} u_{jk})^{q_{kij}}}{q_{kij}!} = \qty[\prod_{i \le j}^{2N} \frac{C_{ij}^{q_{kij}}}{q_{kij}!}] \qty[\prod_{b=1}^{2N} u_{bk}^{\epsilon_{bk}}] \]

and by generalizing to all values of $k$, we see that

(12)
\[\prod_{i \le j}^{2N} \prod_{k=1}^d \frac{(C_{ij} u_{ik} u_{jk})^{q_{kij}}}{q_{kij}!} = \qty[\prod_{i \le j}^{2N} \prod_{k=1}^d \frac{C_{ij}^{q_{kij}}}{q_{kij}!}] \qty[\prod_{i=1}^{2N} \prod_{k=1}^d u_{ik}^{\epsilon_{ik}}]. \]

Now, we need to take derivatives with respect to the parameters $u_{ik}$.

(13)
\[\begin{aligned} \qty[\prod_{i=1}^{2N} \prod_{k=1}^d \pdv[p_{ik}]{u_{ik}}] \qty[\prod_{i=1}^{2N} \prod_{k=1}^d u_{ik}^{\epsilon_{ik}}] &= \prod_{i=1}^{2N} \prod_{k=1}^d \pdv[p_{ik}]{u_{ik}} u_{ik}^{\epsilon_{ik}} \\ &= \prod_{i=1}^{2N} \prod_{k=1}^d \frac{\qty(\epsilon_{ik})!} {\qty(\epsilon_{ik} - p_{ik})!} u_{ik}^{\epsilon_{ik} - p_{ik}} \end{aligned} \]

Finally, we need to set $\uvu = \uvo$. Observe that the preceding expression is nonzero only if $\epsilon_{ik} = p_{ik}$, that is, if

(14)
\[q_{kii} + \sum_{j=1}^{2N} q_{kij} = p_{ik} \]

for all $i = 1, \dots, 2N$ and $k = 1, \dots, d$. In this case, each factor in the product becomes $0^0 = 1$. Assuming that this condition is satisfied, the expression (13) simply reduces to

(15)
\[\prod_{i=1}^{2N} \prod_{k=1}^d p_{ik}! \]

This leads us naturally to consider the following combinatorial problem:

Definition: Let $p$ be a $2N \times d$ matrix with nonnegative integer entries. We denote by $\Omega(p)$ the set of all $d$-tuples of symmetric $2N \times 2N$, again having nonnegative integer entries, satisfying the condition

(16)
\[q_{kii} + \sum_{j=1}^{2N} q_{kij} = p_{ik} \]

for all $i = 1, \dots, 2N$ and $k = 1, \dots, d$.

We will return to solving this problem later. For now, let us assume that we know how to compute the set $\Omega(p)$. The overlap matrix element of two ECG basis functions is then given by

(17)
\[\begin{aligned} \braket{\Phi_1}{\Phi_2} &= \qty[ \prod_{i=1}^N \prod_{k=1}^d \pdv[(p_1)_{ik}]{(t_1)_{ik}} ] \qty[ \prod_{i=1}^N \prod_{k=1}^d \pdv[(p_2)_{ik}]{(t_2)_{ik}} ] \eval{\braket{G_1}{G_2}}_{\uvt_1 = \uvt_2 = \uvo} \\ &= \qty[\frac{(2\pi)^N}{\det A}]^{d/2} \qty[\prod_{i=1}^{N} \prod_{k=1}^d (p_1)_{ik}!] \qty[\prod_{i=1}^{N} \prod_{k=1}^d (p_2)_{ik}!] \times \\ &\pe \sum_{q \in \Omega(\uvp_1, \uvp_2)} \qty[ \prod_{i=1}^{2N} \prod_{j=1}^{2N} \prod_{k=1}^d \frac{1}{q_{ijk}!} \qty(\frac{1}{2} B_{ij})^{q_{ijk}}] \end{aligned} \]
(18)
\[\begin{aligned} \pdv{x_{\iota\kappa}} G_2(\uvx) &= \pdv{x_{\iota\kappa}} \exp( -\frac{1}{2} \uvx^T \! A_2 \uvx + \uvt_2^T \uvx) \\ &= \pdv{x_{\iota\kappa}} \exp( -\frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^d (A_2)_{ij} x_{ik} x_{jk} + \sum_{i=1}^N \sum_{k=1}^d (t_2)_{ik} x_{ik}) \\ &= G_2(\uvx) \pdv{x_{\iota\kappa}} \qty[ -\frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^d (A_2)_{ij} x_{ik} x_{jk} + \sum_{i=1}^N \sum_{k=1}^d (t_2)_{ik} x_{ik} ] \\ &= G_2(\uvx) \qty[ -\frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^d (A_2)_{ij} \pdv{(x_{ik} x_{jk})}{x_{\iota\kappa}} + \sum_{i=1}^N \sum_{k=1}^d (t_2)_{ik} \pdv{x_{ik}}{x_{\iota\kappa}} ] \\ &= G_2(\uvx) \qty[ -\frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^d (A_2)_{ij} \qty(x_{ik} \delta_{j\iota} + \delta_{i\iota} x_{jk}) \delta_{k\kappa} + \sum_{i=1}^N \sum_{k=1}^d (t_2)_{ik} \delta_{i\iota} \delta_{k\kappa} ] \\ &= G_2(\uvx) \qty[ -\frac{1}{2} \sum_{i=1}^N (A_2)_{i\iota} x_{i\kappa} -\frac{1}{2} \sum_{j=1}^N (A_2)_{\iota j} x_{j\kappa} + (t_2)_{\iota\kappa} ] \\ &= G_2(\uvx) \qty[ -\frac{1}{2} \sum_{i=1}^N (A_2)_{i\iota} x_{i\kappa} -\frac{1}{2} \sum_{j=1}^N (A_2)_{\iota j} x_{j\kappa} + (t_2)_{\iota\kappa} ] \\ &= G_2(\uvx) \qty[ -\sum_{i=1}^N (A_2^\text{sym})_{i\iota} x_{i\kappa} + (t_2)_{\iota\kappa} ] \end{aligned} \]
(19)
\[\begin{aligned} \pdv[2]{x_{\iota\kappa}} G_2(\uvx) &= \pdv{G_2}{x_{\iota\kappa}} \qty[ -\sum_{i=1}^N (A_2^\text{sym})_{i\iota} x_{i\kappa} + (t_2)_{\iota\kappa} ] + G_2(\uvx) \qty[ -\sum_{i=1}^N (A_2^\text{sym})_{i\iota} \pdv{x_{i\kappa}}{x_{\iota\kappa}} ] \\ &= G_2(\uvx) \qty[ -\sum_{i=1}^N (A_2^\text{sym})_{i\iota} x_{i\kappa} + (t_2)_{\iota\kappa} ]^2 - (A_2)_{\iota\iota} G_2(\uvx) \\ &= \qty[ \qty(\sum_{i=1}^N (A_2^\text{sym})_{i\iota} x_{i\kappa})^2 - 2 (t_2)_{\iota\kappa} \sum_{i=1}^N (A_2^\text{sym})_{i\iota} x_{i\kappa} + (t_2)_{\iota\kappa}^2 - (A_2)_{\iota\iota} ] G_2(\uvx) \\ &= \Bigg[ \sum_{i=1}^N \sum_{j=1}^N (A_2^\text{sym})_{i\iota} (A_2^\text{sym})_{j\iota} \pdv{(t_2)_{i\kappa}} \pdv{(t_2)_{j\kappa}} \\ &\pe \phantom{\Bigg[} - 2 (t_2)_{\iota\kappa} \sum_{i=1}^N (A_2^\text{sym})_{i\iota} \pdv{(t_2)_{i\kappa}} + (t_2)_{\iota\kappa}^2 - (A_2)_{\iota\iota} \Bigg] G_2(\uvx) \end{aligned} \]