Path Integral Molecular Dynamics Methods

Abstract

We present the aspects of the path integral molecular dynamics (PIMD) method relative to (a) its theoretical fundamental principles, (b) its applicability to model quantum systems and (c) its implementation as a simulation tool. The PIMD method is based on the discretized path integral representation of quantum mechanics. In this representation, a quantum particle is isomorphic to a closed polymer chain. The problem of the indistinguishability of quantum particles is tackled with a non-local exchange potential. When the exact density matrix of the quantum particles is used, the exchange potential is exact. We use a high temperature approximation to the density matrix leading to an approximate form of the exchange potential. This quantum molecular dynamics method allows the simulation of collections of quantum particles at finite temperature. Our algorithm can be made to scale linearly with the number of quantum states on which the density matrix is projected. Therefore, it can be optimized to run efficiently on parallel computers. We apply the PIMD method to the electron plasma in 3-dimension. The kinetic and potential energies are calculated and compared with results for similar systems simulated with a variational Monte Carlo method. Both results show good agreements with each other at all the densities studied. The method is then use to model the thermodynamic behavior of a simple alkali metal. In these simulations, ions and valence electrons are treated as classical and quantum particles, respectively. The simple metal undergoes a phase transformation upon heating. Furthermore, to demonstrate the richness of behaviors that can be studied with the PIMD method, we also report on the metal to insulator transition in a hydrogenoid lattice. Finally, in previous studies of alkali metals, electrons interacted with ions via local pseudo-potentials, an extension of the method to modeling electrons in non-local pseudo-potentials is also presented with applications.

Appendices

Appendix 1: Free Particle Propagator

In this Appendix, we will prove equation (2.16)

$$\rho_{0} \left( {r_{n} ,r_{n + 1} ; \epsilon} \right) = \left( {\frac{m}{{2\pi \epsilon\hbar^{2} }}} \right)^{3/2} \exp\left( { - \frac{{m\left( {r_{n} - r_{n + 1} } \right)^{2} }}{{2 \epsilon\hbar^{2} }}} \right)$$

(2.153)

We assume that every state is defined by a set of plane waves. The free particle propagator in the position representation can be written as

$$\begin{aligned} \rho_{0} (r_{i} ,r_{j} ; \epsilon) & = \left\langle {r_{i} \left| {e^{{ - \epsilon\widehat{\rm T}}} } \right|r_{j} } \right\rangle \\ & = \int {dp\left\langle {r_{i} \left| p \right.} \right\rangle \left\langle {p\left| {e^{{ - \epsilon\widehat{p}^{2} /2m}} } \right|r_{j} } \right\rangle } \\ & = \int {dp\left\langle {r_{i} \left| p \right.} \right\rangle \left\langle {p\left| {r_{j} } \right.} \right\rangle }^{{e^{{ - \epsilon\widehat{p}^{2} /2m}} }} \\ \end{aligned}$$

(2.154)

We used a completeness of momentum states, $1 = \int {dp\left| {\left. p \right\rangle \left\langle p \right.} \right|}$, in (2.154). If we use a plane wave basis $\left\langle {r_{i} |p} \right\rangle = \frac{1}{{\left( {2\pi \hbar } \right)^{3/2} }}{ \exp }\left( {ip \cdot r_{i} /\hbar } \right)$, (2.154) becomes

$$\begin{aligned} \rho_{0} \left( {r_{i} , r_{j} ; \epsilon} \right) & = \smallint \frac{dp}{{\left( {2\pi \hbar } \right)^{3} }}e^{{ip \cdot \left( {r_{i} - r_{j} } \right)/\hbar }} e^{{ - \epsilon p^{2} /2m}} \\ & = \frac{4\pi }{{\left( {2\pi \hbar } \right)^{3} }}\mathop \smallint \limits_{0}^{\infty } dp\, p \frac{{\sin \left( {pr_{ij} /\hbar } \right)}}{{r_{ij} /\hbar }}e^{{ - \epsilon p^{2} /2m}} \\ \end{aligned}$$

(2.155)

where $r_{ij} = \left| {r_{i} - r_{j} } \right|$. From an integral table, we have

$$\int\limits_{0}^{\infty } {x\,\sin \left( {tx} \right) e^{{ - ax^{2} }} dx = \frac{1}{4a}\sqrt {\frac{\pi }{a}} t e^{{ - t^{2} /4a}} }$$

Thus if we set ${\text{a}} =\, \epsilon/2m\;{\text{and}}\;t = r_{ij} /\hbar$, we obtain (2.153).

Appendix 2: Exchange Force Calculation

Here we calculate the exchange force resulting from the effective exchange potential in the Hamiltonian given by (2.146)

$$f_{i,exch}^{\left( k \right)} = \frac{1}{\beta }\mathop \sum \limits_{\mu = 1}^{P} \mathop \sum \limits_{v = 1}^{P} \frac{{\theta_{\mu v}^{ + } /P^{*} }}{{\det\left( {E^{{\left( {\mu ,v} \right)}} } \right)}}\mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} \mathop \sum \limits_{q = 1}^{{N^{{\prime }} }} \frac{{\partial \left( {E^{{\left( {\mu ,v} \right)}} } \right)_{pq} }}{{\partial r_{i}^{\left( k \right)} }}\left( {B^{{\left( {\mu ,v} \right)}} } \right)_{pq}$$

(2.156)

where

$$\begin{aligned} \left( {E^{{\left( {\mu ,v} \right)}} } \right)_{pq} & = \exp\left\{ { - C_{0} \left( {\left( {r_{p}^{\left( \mu \right)} \, - \, r_{q}^{\left( v \right)} } \right)^{2} - \left( {r_{p}^{\left( \mu \right)} - r_{p}^{\left( v \right)} } \right)^{2} } \right)} \right\} \\ & = \exp\left\{ {C_{0} \left( {\left( {r_{p}^{\left( v \right)} } \right)^{2} - \left( {r_{q}^{\left( v \right)} } \right)^{2} \, + \, 2r_{p}^{\left( \mu \right)} \cdot r_{q}^{\left( v \right)} - 2r_{p}^{\left( \mu \right)} \cdot r_{p}^{\left( v \right)} } \right)} \right\} \\ \end{aligned}$$

(2.157)

with $C_{0} = \frac{Pm}{{2\beta \hbar^{2} }}. \left( {B^{{\left( {\mu ,v} \right)}} } \right)_{pq}$ is a cofactor of a matrix element $\left( {E^{{\left( {\mu ,v} \right)}} } \right)_{pq}$. Using (2.157), we obtain

$$\begin{aligned} \frac{{\partial \left( {E^{{\left( {\mu ,v} \right)}} } \right)_{pq} }}{{\partial r_{i}^{\left( k \right)} }} &= 2C_{0} \left( {r_{p}^{\left( v \right)} \delta_{i,p} \delta_{k,v} - r_{q}^{\left( v \right)} \delta_{i,q} \delta_{k,v} + r_{q}^{\left( v \right)} \delta_{i,p} \delta_{k,\mu } + r_{p}^{\left( \mu \right)} \delta_{i,q} \delta_{k,v} - r_{p}^{\left( v \right)} \delta_{i,p} \delta_{k,\mu } - r_{p}^{\left( \mu \right)} \delta_{i,p} \delta_{k,v} } \right)\\ & \quad \left( {E^{{\left( {\mu ,v} \right)}} } \right)_{pq} \end{aligned}$$

(2.158)

The first two terms of the left-hand side of (2.158) with (2.156) become

$$\begin{aligned} f_{i,exch}^{\left( k \right),1st,2nd} & = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{\mu = 1}^{P} \mathop \sum \limits_{v = 1}^{P} \frac{{\frac{{\theta_{\mu v}^{ + } }}{{P^{*} }}}}{{\det\left( {E^{{\left( {\mu ,v} \right)}} } \right)}}\mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} \mathop \sum \limits_{q = 1}^{{N^{{\prime }} }} \left( {r_{p}^{\left( v \right)} \delta_{i,p} \delta_{k,v} - r_{q}^{\left( v \right)} \delta_{i,q} \delta_{k,v} } \right)E_{pq}^{{\left( {\mu ,v} \right)}} B_{pq}^{{\left( {\mu ,v} \right)}} \\ & = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{\mu = 1}^{P} \frac{{\frac{{\theta_{\mu k}^{ + } }}{{P^{*} }}}}{{\det\left( {E^{{\left( {\mu ,k} \right)}} } \right)}}\left( {\mathop \sum \limits_{q = 1}^{{N^{{\prime }} }} r_{i}^{\left( k \right)} E_{iq}^{{\left( {\mu ,k} \right)}} B_{iq}^{{\left( {\mu ,k} \right)}} - \mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} r_{i}^{\left( k \right)} E_{pi}^{{\left( {\mu ,k} \right)}} B_{pi}^{{\left( {\mu ,k} \right)}} } \right) = 0 \\ \end{aligned}$$

(2.159)

In (2.159), we used the relation ${ \det }\left( {E^{{\left( {k,\mu } \right)}} } \right) = { \det }\left( {E^{{\left( {\mu ,k} \right)}} } \right)$ and the matrix algebra

$$\det A = \mathop \sum \limits_{i = 1}^{N} a_{ij} A_{ij} = \mathop \sum \limits_{j = 1}^{N} a_{ij} A_{ij}$$

From the third and the fifth terms of equation (2.158), we obtain:

$$\begin{aligned} f_{i,exch}^{\left( k \right),3rd,5th} & = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{\mu = 1}^{P} \mathop \sum \limits_{v = 1}^{P} \frac{{\theta_{\mu v}^{ + } /P^{*} }}{{\det\left( {E^{{\left( {\mu ,v} \right)}} } \right)}}\mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} \mathop \sum \limits_{q = 1}^{{N^{{\prime }} }} \left( {r_{q}^{\left( v \right)} \delta_{i,p} \delta_{k,\mu } - r_{p}^{\left( v \right)} \delta_{i,p} \delta_{k,\mu } } \right)E_{pq}^{{\left( {\mu ,v} \right)}} B_{pq}^{{\left( {\mu ,v} \right)}} \\ & = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{v = 1}^{P} \frac{{\theta_{kv}^{ + } /P^{*} }}{{\det\left( {E^{{\left( {k,v} \right)}} } \right)}}\mathop \sum \limits_{q = 1}^{{N^{{\prime }} }} \left( {r_{q}^{\left( v \right)} - r_{i}^{\left( v \right)} } \right)E_{iq}^{{\left( {k,v} \right)}} B_{iq}^{{\left( {k,v} \right)}} \\ \end{aligned}$$

(2.160)

Similarly from the fourth and sixth terms, we get

$$\begin{aligned} f_{i,exch}^{\left( k \right),4th,6th} & = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{\mu = 1}^{P} \mathop \sum \limits_{v = 1}^{P} \frac{{\theta_{\mu v}^{ + } /P^{*} }}{{\det\left( {E^{{\left( {\mu ,v} \right)}} } \right)}}\mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} \mathop \sum \limits_{q = 1}^{{N^{{\prime }} }} \left( {r_{p}^{\left( \mu \right)} \delta_{i,q} \delta_{k,v} - r_{p}^{\left( \mu \right)} \delta_{i,p} \delta_{k,v} } \right)E_{pq}^{{\left( {\mu ,v} \right)}} B_{pq}^{{\left( {\mu ,v} \right)}} \\ & = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{\mu = 1}^{P} \frac{{\theta_{\mu k}^{ + } /P^{*} }}{{\det\left( {E^{{\left( {\mu ,k} \right)}} } \right)}}\left( {\mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} r_{p}^{\left( \mu \right)} E_{pi}^{{\left( {\mu ,k} \right)}} B_{pi}^{{\left( {\mu ,k} \right)}} - \mathop \sum \limits_{q = 1}^{{N^{{\prime }} }} r_{i}^{\left( \mu \right)} E_{iq}^{{\left( {\mu ,k} \right)}} B_{iq}^{{\left( {\mu ,k} \right)}} } \right) \\ & = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{\mu = 1}^{P} \frac{{\theta_{\mu k}^{ + } /P^{*} }}{{\det\left( {E^{{\left( {\mu ,k} \right)}} } \right)}}\left( {\mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} r_{p}^{\left( \mu \right)} E_{pi}^{{\left( {\mu ,k} \right)}} B_{pi}^{{\left( {\mu ,k} \right)}} - \mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} r_{i}^{\left( \mu \right)} E_{pi}^{{\left( {\mu ,k} \right)}} B_{pi}^{{\left( {\mu ,k} \right)}} } \right) \\ & = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{\mu = 1}^{P} \frac{{\theta_{\mu k}^{ + } /P^{*} }}{{\det\left( {E^{{\left( {\mu ,k} \right)}} } \right)}}\mathop \sum \limits_{p = 1}^{{N^{{\prime }} }} \left( {r_{p}^{\left( \mu \right)} - r_{i}^{\left( \mu \right)} } \right)E_{pi}^{{\left( {\mu ,k} \right)}} B_{pi}^{{\left( {\mu ,k} \right)}} \\ \end{aligned}$$

(2.161)

From the above equations, the exchange force felt by the kth bead of the ith electron becomes

$$f_{i,exch}^{\left( k \right)} = \frac{Pm}{{\beta^{2} \hbar^{2} }}\mathop \sum \limits_{\mu = 1}^{P} \frac{{\theta_{k\mu }^{ + } /P^{*} }}{{\det\left( {E^{{\left( {k,\mu } \right)}} } \right)}}\left\{ {\det\left( {f_{i}^{{\left( {k,\mu } \right)}} } \right) + \det\left( {G_{i}^{{\left( {\mu ,k} \right)}} } \right)} \right\}$$

(2.162)

where the elements of matrix $f_{i}^{{\left( {k,v} \right)}}$ and $G_{i}^{{\left( {v,k} \right)}}$ are defined as

$$\left( {F_{i}^{{\left( {k,\mu } \right)}} } \right)_{pq} = \left\{ {\begin{array}{*{20}l} {\left( {r_{q}^{\left( \mu \right)} - r_{i}^{\left( \mu \right)} } \right)\left( {E^{{\left( {k,\mu } \right)}} } \right)_{pq} } \hfill & {if\,p = 1} \hfill \\ {\left( {E^{{\left( {k,\mu } \right)}} } \right)_{pq} } \hfill & {if\,p \ne 1} \hfill \\ \end{array} } \right.$$

(2.163)

and

$$\left( {G_{i}^{{\left( {\mu ,k} \right)}} } \right)_{pq} = \left\{ {\begin{array}{*{20}l} {\left( {r_{p}^{\left( \mu \right)} - r_{i}^{\left( \mu \right)} } \right)\left( {E^{{\left( {\mu ,k} \right)}} } \right)_{pq} } \hfill & {if\,q = i} \hfill \\ {\left( {E^{{\left( {\mu ,k} \right)}} } \right)_{pq} } \hfill & {if\,q \ne i} \hfill \\ \end{array} } \right.$$

(2.164)

Appendix 3: Derivation of the Force on an Electron in a Non-local Pseudo-potential

We start with the expression for the effective potential of the electron necklace in a non-local pseudo-potential given by (2.125) which is repeated below

$$\begin{aligned} V_{eff} & = - \frac{1}{\beta }\mathop \sum \limits_{n = 1}^{P} L{\text{n}}\left\{ {e^{{ - \frac{\beta }{P}V_{loc} }} e^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} \times {\mathfrak{F}}_{{\frac{1}{2}}} \left( {2\beta Cr_{n} r_{n + 1} } \right)\left[ {e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} - e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} } \right]} \right. \\ & \quad \quad \left. { \, + e^{{ - \frac{\beta }{P}\left[ {V_{loc} + V_{1} \left( {r_{n} } \right)} \right]}} e^{{ - \beta C\left( {\vec{r}_{n} - \vec{r}_{n + 1} } \right)^{2} }} } \right\} \\ \end{aligned}$$

(2.165)

The total effective force on the electron is the vector sum of all the forces acting on the nodes. The force on the nth node of the electron is given by:

$$\vec{F}_{n } = - \frac{{\partial V_{eff} }}{{\partial \vec{r}_{n} }}$$

(2.166)

Using $x = 2\beta Cr_{n} r_{n + 1}$ and absorbing $V_{loc}$ in the angular momentum, dependent potentials will help us to simplify the bookkeeping. The force on the nth node is the given by:

$$\begin{aligned} \vec{F}_{n} = & \frac{1}{ < \ldots > }\left\{ { - 2C\left( {r_{n} - r_{n + 1} } \right)\frac{{\vec{r}_{n} }}{{r_{n} }}e^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} \times {\mathfrak{F}}_{{\frac{1}{2}}} \left( \chi \right)\left[ {e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} - e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} } \right]} \right. \\ & \quad + 2Cr_{n + 1} \frac{{\vec{r}_{n} }}{{r_{n} }}\frac{{e^{ - \chi } }}{\chi }\left[ {\cosh \chi - \sinh \chi \left( {1 + \frac{1}{\chi }} \right)} \right] \\ & \quad \times e^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} \left[ {e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} - e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} } \right] \\ & \quad - \frac{{\vec{r}_{n} }}{{r_{n} }}e^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} {\mathfrak{F}}_{{\frac{1}{2}}} \left( \chi \right)\left[ {\frac{1}{P}e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} \frac{{\partial V_{0} \left( {r_{n} } \right)}}{{\partial r_{n} }} - \frac{1}{P}e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} \frac{{\partial V_{1} \left( {r_{n} } \right)}}{{\partial r_{n} }}} \right] \\ & \quad \left. { + \, e^{{ - \beta C\left( {\vec{r}_{n} - \vec{r}_{n + 1} } \right)^{2} }} e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} \times \left[ { - 2C\left( {\vec{r}_{n} - \vec{r}_{n + 1} } \right) - \frac{1}{P}\frac{{\partial V_{1} \left( {r_{n} } \right)}}{{\partial r_{n} }}} \right]} \right\} \\ \end{aligned}$$

(2.167)

Where < … > is the argument of the Logarithmic function in the expression of $V_{eff}$. We would like to note that in deriving the above expression the modified Bessel function was used in its integral form [62]:

$$I_{\tau } \left( \chi \right) = \frac{{\left( {\frac{\chi }{2}} \right)^{\tau } }}{{\Gamma\left( {\tau + \frac{1}{2}} \right)\Gamma\left( {\frac{1}{2}} \right)}}\mathop \smallint \limits_{ - 1}^{1} \left( {1 - y^{2} } \right)^{{\tau - \frac{1}{2}}} e^{ \pm \chi y} dy$$

where $\Gamma$ is the familiar Gamma function. In our case this form could be simplified to become: ${\mathfrak{F}}_{{\frac{1}{2}}} \left( \chi \right) = \frac{{e^{ - \chi } }}{\chi }\sin {\text{h}}\chi$ where we use the fact that $\Gamma \left( {1/2} \right) = \sqrt \pi \, and\,\Gamma \left( 1 \right) = 1$.

The force on the $\left( {n + 1} \right){\text{th}}$ node is calculated in a similar manner and given by:

$$\begin{aligned} \vec{F}_{n + 1} = & \frac{1}{ < \ldots > }\left\{ {2C\left( {r_{n} - r_{n + 1} } \right)\frac{{\vec{r}_{n + 1} }}{{r_{n + 1} }}e^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} \times {\mathfrak{F}}_{{\frac{1}{2}}} \left( \chi \right)\left[ {e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} - e^{{ - \frac{\beta }{P}V_{1} r_{n} }} } \right]} \right. \\ & \quad + 2Ce^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} \frac{{\vec{r}_{n + 1} }}{{r_{n + 1} }}r_{n} \frac{{e^{ - \chi } }}{\chi }\left[ {\cos \text{h}\chi - \sin \text{h}\chi \left( {1 + \frac{1}{\chi }} \right)} \right]\left[ {e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} - e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} } \right] \\ & \quad \left. { + 2Ce^{{ - \beta C\left( {\vec{r}_{n} - \vec{r}_{n + 1} } \right)^{2} }} e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} \left( {\vec{r}_{n} - \vec{r}_{n + 1} } \right)} \right\} \\ \end{aligned}$$

(2.168)

Appendix 4: Exchange Kinetic Energy Estimator for N-Electron System

In this section, we will calculate the exchange kinetic estimator, ${\langle{KE_{exch}} \rangle}$, which is given

$$\begin{aligned} KE_{exch} & = {\left\langle{\frac{\partial }{\partial \beta }\left( {\beta V_{eff}^{exch} } \right) }\right\rangle}\\ & = {\left\langle{\frac{\partial }{\partial \beta }\mathop \sum \limits_{s = up}^{down} \mathop \sum \limits_{k = 1}^{P} \mathop \sum \limits_{l = 1}^{P} \frac{1}{{P^{*} }}\ln \left( {\det\left( {E^{{\left( {k,l} \right)}} } \right)} \right)\theta_{kls}^{ + } }\right\rangle}\\ & = {\left\langle{\mathop \sum \limits_{s = up}^{down} \frac{1}{{P^{*} }}\mathop \sum \limits_{k = 1}^{P} \mathop \sum \limits_{l = 1}^{P} \frac{1}{{\det\left( {E^{{\left( {k,l} \right)}} } \right)}}\frac{\partial }{\partial \beta }\det\left( {E^{{\left( {k,l} \right)}} } \right)\theta_{kls}^{ + } }\right\rangle}\\ \end{aligned}$$

(2.169)

In order to differentiate a determinant, we use the following matrix algebra: if a $N \times N$ square matrix A is function of X, we have

$$\begin{aligned} \hfill \\ \begin{array}{*{20}c} {\det A = \sum\limits_{i = 1}^{N} {a_{ij} } A_{ij} = \sum\limits_{j = 1}^{N} {a_{ij} } A_{ij} } \\ { (6.143)} \\ {\frac{\partial }{\partial X}\det A = \sum\limits_{i = 1}^{N} {\sum\limits_{j = 1}^{N} {\frac{{\partial a_{ij} }}{\partial X}} } A_{ij} } \\ \end{array} \hfill \\ \end{aligned}$$

(2.170)

where a _ij is an element of the matrix A and A _ij is a cofactor of the element a _ij . Thus if we differentiate ${ \det }\left( {E^{{\left( {k,l} \right)}} } \right)$ with respect to $\beta$, we have

$$\begin{aligned} \frac{\partial }{\partial \beta }\det\left( {E^{{\left( {k,l} \right)}} } \right) & = \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} \frac{\partial }{\partial \beta }\left( {E^{{\left( {k,l} \right)}} } \right)_{ij} \, \cdot \,A_{ij} \\ & = \mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} \frac{\partial }{\partial \beta }\left[ {\exp\left\{ { - \frac{Pm}{{2\beta \hbar^{2} }}\left( {\left( {r_{i}^{\left( k \right)} - r_{j}^{\left( l \right)} } \right)^{2} - \left( {r_{i}^{\left( k \right)} - r_{i}^{\left( l \right)} } \right)^{2} } \right)} \right\}} \right] \cdot A_{ij} \\ & = \frac{Pm}{{2\beta^{2} \hbar^{2} }}\mathop \sum \limits_{i = 1}^{N} \mathop \sum \limits_{j = 1}^{N} \left( {\left( {r_{i}^{\left( k \right)} - r_{j}^{\left( l \right)} } \right)^{2} - \left( {r_{i}^{\left( k \right)} - r_{i}^{\left( l \right)} } \right)^{2} } \right)\left( {E^{{\left( {k,l} \right)}} } \right)_{ij} \, \cdot \, A_{ij} \\ \end{aligned}$$

(2.171)

If we apply (2.170) to the left-hand side of the last term of (2.171), we can write that equation in a simpler form:

$$\frac{\partial }{\partial \beta }\det\left( {E^{{\left( {k,l} \right)}} } \right) = \frac{Pm}{{2\beta^{2} \hbar^{2} }}\mathop \sum \limits_{i = 1}^{N} \det\left( {H_{i}^{{\left( {k,l} \right)}} } \right)$$

(2.172)

where $H_{i}^{{\left( {k,l} \right)}}$ is an $N \times N$ matrix and its element $\left( {H_{i}^{{\left( {k,l} \right)}} } \right)_{st}$ is given by

$$\begin{aligned} \left( {H_{i}^{{\left( {k,l} \right)}} } \right)_{st} = & \left\{ {\begin{array}{*{20}l} {L_{st}^{{\left( {k,l} \right)}} \exp\left( { - \frac{Pm}{{2\beta^{2} \hbar^{2} }}L_{st}^{{\left( {k,l} \right)}} } \right)} \hfill & {if\,i = t} \hfill \\ {\exp\left( { - \frac{Pm}{{2\beta^{2} \hbar^{2} }}L_{st}^{{\left( {k,l} \right)}} } \right)} \hfill & {if\,i \ne t} \hfill \\ \end{array} } \right. \\ & {(6.147)} \end{aligned}$$

and

$$L_{st}^{{\left( {k,l} \right)}} = \left( {r_{s}^{\left( k \right)} - r_{t}^{\left( l \right)} } \right)^{2} - \left( {r_{s}^{\left( k \right)} - r_{s}^{\left( l \right)} } \right)^{2}$$

From (2.170) and (2.172), the exchange kinetic estimator can be written as

$${\langle {KE_{exch}} \rangle} = {\left\langle{\mathop \sum \limits_{s = up}^{down} \frac{1}{{P^{*} }}\mathop \sum \limits_{k = 1}^{P} \mathop \sum \limits_{l = 1}^{P} \frac{1}{{\det\left( {E^{{\left( {k,l} \right)}} } \right)}}\mathop \sum \limits_{i = 1}^{N} \frac{Pm}{{2\beta^{2} \hbar^{2} }}\det\left( {H_{i}^{{\left( {k,l} \right)}} } \right)\theta_{kls}^{ + }}\right\rangle}$$

(2.173)

Appendix 5: Energy Estimator for Electron in Non-local Pseudo-potential

The energy could be calculated easily using the canonical ensemble. In this case the energy is given by:

$$\langle {E} \rangle= - \frac{1}{Z}\frac{\partial Z}{\partial \beta }$$

(2.174)

Using the partition function given by (2.124) the ensemble average of the energy is:

$${\left\langle {E} \right \rangle}= \left( {\frac{3P}{2\beta }} \right) - {\left\langle {\mathop \sum \limits_{n = 1}^{P} \frac{1}{\left\langle \ldots \right\rangle }\frac{\partial \left\langle \ldots \right\rangle }{\partial \beta } }\right \rangle}$$

(2.175)

Where < … > is the argument of the Ln function in the effective potential of an electron in the non-local pseudo-potential. For the sodium ion, the average energy is then given by the following equation:

$$\begin{aligned} \langle E\rangle & = \left( {\frac{3P}{2\beta }} \right) - \left\langle {\sum\limits_{n = 1}^{P} {\frac{1}{ < \ldots > }} \left\{ {C\left( {r_{n} - r_{n + 1} } \right)^{2} \times e^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} {\mathfrak{F}}_{{\frac{1}{2}}} \left( \chi \right)\left[ {e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} - e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} } \right]} \right.} \right. \\ & \quad + e^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} {\mathfrak{F}}_{{\frac{1}{2}}} \left( \chi \right)\left[ {\frac{{V_{1} \left( {r_{n} } \right)}}{P}e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} - \frac{{V_{0} \left( {r_{n} } \right)}}{P}e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} } \right] \\ & \quad - e^{{ - \beta C\left( {r_{n} - r_{n + 1} } \right)^{2} }} \left[ {e^{{ - \frac{\beta }{P}V_{0} \left( {r_{n} } \right)}} - e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} } \right]\left( {\frac{\chi }{\beta }} \right)\left[ {\frac{{e^{ - 2\chi } }}{\chi }\left( {1 + \frac{1}{2\chi }} \right) - \frac{1}{{2\chi^{2} }}} \right] \\ & \left. {\quad \left. { + \, C\left( {\vec{r}_{n} - \vec{r}_{n + 1} } \right)^{2} e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} e^{{ - \beta C\left( {\vec{r}_{n} - \vec{r}_{n + 1} } \right)^{2} }} - \frac{{V_{1} \left( {r_{n} } \right)}}{P}e^{{ - \frac{\beta }{P}V_{1} \left( {r_{n} } \right)}} e^{{ - \beta C\left( {\vec{r}_{n} - \vec{r}_{n + 1} } \right)^{2} }} } \right\}} \right\rangle \\ \end{aligned}$$

(2.176)