**Quantal Cumulant Mechanics as Extended Ehrenfest Theorem**

Yasuteru Shigeta

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/53703

### **1. Introduction**

Since Schrödinger proposed wave mechanics for quantum phenomena in 1926 [1-4], referred as Schrödinger equation named after his name, this equation has been applied to atom-mol‐ ecules, condensed matter, particle, and elementary particle physics and succeeded to repro‐ duce various experiments. Although the Schrödinger equation is in principle the differential equation and difficult to solve, by introducing trial wave functions it is reduced to matrix equations on the basis of the variational principle. The accuracy of the approximate Schrö‐ dinger equation depends strongly on the quality of the trial wave function. He also derived the time-dependent Schrödinger equation by imposing the time-energy correspondence. This extension opened to describe time-dependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schrö‐ dinger equation approximately is extensively explored, yet.

In contrast to the time-dependent wave mechanics, Heisenberg developed the equations of motion (EOM) derived for time-dependent operator rather than wave function [5]. This equation is now referred as the Heisenberg' EOM. This equation is exactly equivalent to the time-dependent Schrödinger equation so that the trials to solve the Heisenberg' EOM rather than Schrödinger one were also done for long time. For example, the Dyson equation, which is the basic equation in the Green's function theory, is also derived from the Heisenberg' EOM. Various approximate methods were deviced to solve the Dyson equation for nuclear and electronic structures.

In this chapter, we propose a new approximate methodology to solve dynamical properties of given systems on the basis of quantum mechanics starting from the Heisenberg' EOM. First, theoretical background of the method is given for one-dimensional systems and an ex‐ tension to multi-dimensional cases is derived. Then, we show three applications in molecu‐

© 2013 Shigeta; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Shigeta; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

lar physics, i.e. the molecular vibration, the proton transfer reaction, and the quantum structural transition, respectively. Finally, we give conclusion at the last part.

#### **2. Theoretical background**

#### **2.1. Heisenberg' equation of motion and Ehrenfest theorem**

When the Hamiltonian does not explicitly depend on time, by defining time-dependent of an arbitrary operator *A* ^ in the Heisenberg representation as

$$
\hat{A}(t) = e^{-\frac{\hat{t}}{\hbar}\hat{H}t}\hat{A}(0)e^{\frac{\hat{t}}{\hbar}\hat{H}t}.\tag{1}
$$

These equations resemble corresponding Newton' EOMs as

the former. If one approximates the expectation value as

In general, Taylor expansion of the potential energy term,

gives a infinite series of higher-order derivatives, *V* (*m*)

quantum mechanics is existence of higher-order moment.

**2.2. Quantized Hamilton dynamics and quantal cumulant dynamics**

er-powers of coordinate moment operators, *q*

^ <sup>≡</sup> *<sup>A</sup>* ^ <sup>−</sup> *<sup>A</sup>*

( ( )) ( ( ) ) ( )

ferred as the Ehrenfest's theorem.

^ as *δ<sup>A</sup>*

^(*t*)<sup>−</sup> *<sup>q</sup>*

operator of *A*

^*m*(*t*) <sup>≡</sup> (*<sup>q</sup>*

*δq*

( ) ( )

*p t q t m pt V qt*

&

&

=

( ) ( )

= -

( ( )) <sup>1</sup> .

This relationship is so-called Ehrenfest's theorem [6]. A definite difference between Heisen‐ berg' and Nowton' EOMs is that the expectation value of the potential operator appears in

the same structure of the EOM is immediately derived. However, there is no guarantee that this approximation always holds for general cases. Including this approximation is also re‐

( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12 3 1 1 2 3 ˆ ˆˆ ˆ 00 0 0 , 2! 3! *V qt V V qt V q t V q t* =+ + <sup>+</sup> <sup>+</sup><sup>L</sup> (8)

^(*t*) )*<sup>m</sup>* (*m*=2, 3, <sup>⋯</sup>), the Taylor series is rewritten as

( ( ) ) ( ) ( ) ( ( ) ) ( ) 1 1 2 3 2 3 ˆ ˆ ˆˆ ˆˆ , 2! 3! *V qt V qt V qt q t V qt q t* = + d

The first term appears in Eq. (2-7) and the other terms are neglected by the approximation made before. This relation indicates that the difference between classical mechanics and

Ehrenfest' theorem fulfills for the arbitrary wave function. In previous studies, effects of the higher-order moments on dynamics were explored. The most of studies treat second-order

( ( )) ( ( ) ) *V qt V qt* ˆ ˆ » , (7)

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

http://dx.doi.org/10.5772/53703

(0), times expectation values of high‐

^*m*(*t*) (*m*=1, 2, <sup>⋯</sup>). Introducing a fluctuation

 d+ +L (9)

^ and the expectation values of the higher-order central moment

(6)

295

The Heisenberg' equation of motion (EOM) is given as

$$\frac{\partial \hat{A}\left(t\right)}{\partial t} = \frac{1}{i\hbar} \left[\hat{A}\left(t\right), \hat{H}\right]\_{\prime} \tag{2}$$

where *H* ^ is the Hamiltonian operator and *<sup>h</sup>* =2*π*ℏ is the Planck's constant. As an expectation value of *A* ^ with respect to *<sup>ψ</sup>* is expressed as *<sup>A</sup>* ^ <sup>≡</sup> *<sup>ψ</sup>*<sup>|</sup> *<sup>A</sup>* ^ <sup>|</sup>*<sup>ψ</sup>* , the Heisenberg' EOM is rewrit‐ ten as

$$\frac{\partial \left\langle \hat{A} \left( t \right) \right\rangle}{\partial t} = \frac{1}{i\hbar} \left\langle \left[ \hat{A} \left( t \right) , \hat{H} \right] \right\rangle = \frac{1}{i\hbar} \left\langle e^{-\frac{i}{\hbar} \hat{H}t} \left[ \hat{A} , \hat{H} \right] e^{\frac{i}{\hbar} \hat{H}t} \right\rangle. \tag{3}$$

For one-dimensional case, the Hamiltonian operator is expressed as a sum of the kinetic and the potential operator as

$$
\hat{H} = \frac{\hat{p}^2}{2m} + V\left(\hat{q}\right). \tag{4}
$$

The Heisenberg' EOMs for both a coordinate and a momentum are derived as

$$\begin{cases} \frac{\partial \left< \hat{\boldsymbol{q}}(t) \right>}{\partial t} = \frac{\left< \hat{\boldsymbol{p}}(t) \right>}{m} \\ \frac{\partial \left< \hat{\boldsymbol{p}}(t) \right>}{\partial t} = - \left< \boldsymbol{V}^{(1)} \left( \hat{\boldsymbol{q}}(t) \right) \right> \end{cases} \rightarrow \begin{cases} \dot{\boldsymbol{q}}(t) = \frac{\boldsymbol{p}(t)}{m} \\ \dot{\boldsymbol{p}}(t) = - \left< \boldsymbol{V}^{(1)} \left( \hat{\boldsymbol{q}}(t) \right) \right> \end{cases} \tag{5}$$

These equations resemble corresponding Newton' EOMs as

lar physics, i.e. the molecular vibration, the proton transfer reaction, and the quantum

When the Hamiltonian does not explicitly depend on time, by defining time-dependent of

= h h (1)

^ <sup>|</sup>*<sup>ψ</sup>* , the Heisenberg' EOM is rewrit‐

(2)

(5)

in the Heisenberg representation as

( ) ( ) ˆ ˆ ˆ ˆ 0 . *i i Ht Ht*

( ) ( ) <sup>ˆ</sup> <sup>1</sup> ˆ ˆ , , *A t At H*

( ) ( ) ˆ ˆ <sup>ˆ</sup> 1 1 ˆ ˆ ˆ ˆ , ,. *i i A t Ht Ht*

<sup>=</sup> é ù éù <sup>º</sup> ë û ëû ¶

*At H e AH e*

For one-dimensional case, the Hamiltonian operator is expressed as a sum of the kinetic and

( )

( ) ( )

( ) ( ) ( ( )) <sup>1</sup> <sup>1</sup>

<sup>2</sup> <sup>ˆ</sup> <sup>ˆ</sup> <sup>ˆ</sup> . <sup>2</sup> *<sup>p</sup> H Vq m*

. <sup>ˆ</sup> <sup>ˆ</sup> <sup>ˆ</sup>

&

&

*p t pt V qt V qt <sup>t</sup>*

The Heisenberg' EOMs for both a coordinate and a momentum are derived as

( ( ))

*qt pt p t q t t m <sup>m</sup>*

<sup>ì</sup> ¶ <sup>ì</sup> <sup>ï</sup> <sup>=</sup> <sup>ï</sup> <sup>=</sup> ï ï í í ¶ ®

ï ï ¶ = - = - <sup>î</sup> <sup>î</sup> ¶

^ is the Hamiltonian operator and *<sup>h</sup>* =2*π*ℏ is the Planck's constant. As an expectation

^ <sup>≡</sup> *<sup>ψ</sup>*<sup>|</sup> *<sup>A</sup>*

h h h h (3)

= + (4)

¶ <sup>=</sup> é ù ë û ¶ h

*t i*

*At e A e* -

structural transition, respectively. Finally, we give conclusion at the last part.

**2.1. Heisenberg' equation of motion and Ehrenfest theorem**

The Heisenberg' equation of motion (EOM) is given as

with respect to *ψ* is expressed as *A*

( ) ( )

ˆ ˆ

( ) ( )

*ti i* ¶ -

**2. Theoretical background**

^

an arbitrary operator *A*

294 Advances in Quantum Mechanics

where *H*

ten as

value of *A* ^

the potential operator as

$$\begin{aligned} \dot{q}\left(t\right) &= \frac{p\left(t\right)}{m} \\ \dot{p}\left(t\right) &= -V^{\left(1\right)}\left(q\left(t\right)\right) . \end{aligned} \tag{6}$$

This relationship is so-called Ehrenfest's theorem [6]. A definite difference between Heisen‐ berg' and Nowton' EOMs is that the expectation value of the potential operator appears in the former. If one approximates the expectation value as

$$
\left< V\left(\hat{q}\left(t\right)\right) \right> \approx V\left(\left<\hat{q}\left(t\right)\right>\right). \tag{7}
$$

the same structure of the EOM is immediately derived. However, there is no guarantee that this approximation always holds for general cases. Including this approximation is also re‐ ferred as the Ehrenfest's theorem.

In general, Taylor expansion of the potential energy term,

$$\left\langle V\left(\hat{\eta}(t)\right)\right\rangle = V\left(0\right) + V^{(1)}\left(0\right)\left\langle \hat{\eta}(t)\right\rangle + \frac{1}{2!}V^{(2)}\left(0\right)\left\langle \hat{\eta}^2(t)\right\rangle + \frac{1}{3!}V^{(3)}\left(0\right)\left\langle \hat{\eta}^3(t)\right\rangle + \cdots,\tag{8}$$

gives a infinite series of higher-order derivatives, *V* (*m*) (0), times expectation values of high‐ er-powers of coordinate moment operators, *q* ^*m*(*t*) (*m*=1, 2, <sup>⋯</sup>). Introducing a fluctuation operator of *A* ^ as *δA* ^ <sup>≡</sup> *<sup>A</sup>* ^ <sup>−</sup> *<sup>A</sup>* ^ and the expectation values of the higher-order central moment *δq* ^*m*(*t*) <sup>≡</sup> (*<sup>q</sup>* ^(*t*)<sup>−</sup> *<sup>q</sup>* ^(*t*) )*<sup>m</sup>* (*m*=2, 3, <sup>⋯</sup>), the Taylor series is rewritten as

$$
\left\langle V\left(\hat{q}\left(t\right)\right)\right\rangle = V\left(\left\langle\hat{q}\left(t\right)\right\rangle\right) + \frac{1}{2!}V^{(2)}\left(\left\langle\hat{q}\left(t\right)\right\rangle\right)\left\langle\delta\hat{q}^2\left(t\right)\right\rangle + \frac{1}{3!}V^{(3)}\left(\left\langle\hat{q}\left(t\right)\right\rangle\right)\left\langle\delta\hat{q}^3\left(t\right)\right\rangle + \cdots,\tag{9}
$$

The first term appears in Eq. (2-7) and the other terms are neglected by the approximation made before. This relation indicates that the difference between classical mechanics and quantum mechanics is existence of higher-order moment.

#### **2.2. Quantized Hamilton dynamics and quantal cumulant dynamics**

Ehrenfest' theorem fulfills for the arbitrary wave function. In previous studies, effects of the higher-order moments on dynamics were explored. The most of studies treat second-order term with the potential being a series of *q*. For example, Prezhdo and co-workers derived EOMs for three additional moments of *q* ^2(*t*) , *<sup>p</sup>* ^2(*t*) , and (*<sup>q</sup>* ^(*t*)*<sup>p</sup>* ^(*t*)) <sup>s</sup> and solved the EOMs by truncating the potential term up to fourth-order power series. The subscript s represent a symmetric sum of the operator product defined as (*q* ^(*t*)*<sup>p</sup>* ^(*t*)) <sup>s</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> (*q* ^(*t*)*<sup>p</sup>* ^(*t*) <sup>+</sup> *<sup>p</sup>* ^(*t*)*<sup>q</sup>* ^(*t*)) . Judg‐ ing from previous works, this formalism is essentially the same as Gaussian wave packet method. Prezhdo also proposed a correction to the higher-order moments [8]. Nevertheless their formalism could not be applied general potential without any approximation such as the truncation.

( ) ( )

*p t q t m*

% &

( ) ( )

( ) ( )

l

ll

& %

ï = -

l

2,0

l

&

ï ï í = ï ï

ï

î

l

2

( )


2,0

l

¥ ¥

*<sup>V</sup>*˜(*n*,0)

ì ï = ï

*t*

<sup>ï</sup> = - <sup>ï</sup>

&

ll

where *<sup>V</sup>*˜ is second-order "quantal" potential defined as

( ( )) ( ( ) ) ( )

<sup>ï</sup> = - <sup>ï</sup>

1,1

*m t*

*m*

( ( ) ( ))

l

,

2,0

*t tV qt t*

<sup>2</sup> ,

( ( ) ( ))

( )

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

http://dx.doi.org/10.5772/53703

ˆ

*q qt*

=

( ( ) )

2 2 2

p

*aa a a*

*<sup>r</sup> r dr*

l

æ ö ¶ ¶ é ù æ ö ç ÷ = -+ ê ú ç ÷ ç ÷ ¶ ê ú ¶ è ø ë û è ø <sup>ò</sup> (17)

(13)

297

,

2 2,0

*q*

% (14)

( ( ) ) ( )

 l

2

=+ + +L (15)

 l

( ( ) ( ))

l

is the *n*-th derivative of *<sup>V</sup>*˜ with respect to *<sup>q</sup>*. It is easily seen that the quantal potential

2,0 2 4 2,0

It is noteworthy that the first and second terms of above equation corresponds to the first and second terms of Eq. (2-9), on the other hand, the other term are different each other.

Now we here give an expression to the quantal potential that has complicated form like as

<sup>2</sup> exp exp exp , 24 4 *bb b ar br dr a r dr*

2 2

exp exp . 2 2 <sup>2</sup>

¥ -¥

1

pl

2,0 2,0

*q q*

é ù æ ö æ ö é ù - + = -+ + = ê ú ç ÷ ç ÷ ë û ê ú ç ÷ ê ú è ø è ø ë û

ò ò (16)

 l

( ) ( ) ( ) ( )

0,2 2,0 1,1 2,0 2,0 2,0 0,2 1,1 2,0

2 ,

( ( ) ( )) ( ( )) ( )

*V qt t V qt V q*

2,0 <sup>2</sup> <sup>2</sup>

is a finite series with respect to the cumulant by expanding as a Taylor series as

l

in Eq. (2-14). By using the famous formula for the Gaussian integral

the exponential operator appearing in Eq. (2-14) is rewritten as,

2

ˆˆ ˆ ˆ , 2 8 *V qt V qt V qt V qt*

, <sup>ˆ</sup> exp . <sup>2</sup>

æ ö ¶ º = ç ÷ ç ÷ ¶ è ø

*t tV qt t*

( ) ( ) ( )

& %

1,0

*pt V qt t t*

Recently Shigeta and co-workers derived a general expression for the expectation value of an arbitrary operator by means of cumulants rather than moments [9-19]. For one-dimen‐ sional case, the expectation value of a differential arbitrary operator, *A*<sup>s</sup> (*q* ^, *<sup>p</sup>* ^), that consists of the symmetric sum of power series of *q* and *p* is derived as

$$\left\langle A\_s\left(\hat{q}\left(t\right),\hat{p}\left(t\right)\right)\right\rangle = \exp\left(\sum\_{m}\sum\_{0\le l\le m} \frac{\mathcal{I}\_{l,m-l}\left(t\right)}{l!\left(m-l\right)!} \frac{\hat{\circ}^m}{\hat{\circ}q^l\hat{\circ}p^{m-l}}\right) A\left(q,p\right),\tag{10}$$

where we introduced the general expression for the cumulant *λm*,*n*(*t*)≡ (*δq* ^*m*(*t*)*<sup>δ</sup> <sup>p</sup>* ^*n*(*t*))s , in which the subscripts mean *m*-th order and *n*-th order with respect to the coordinate and mo‐ mentum, respectively [20-22]. Using the expression, the expectation value of the potential is

$$\left\langle V\left(\hat{q}\left(t\right)\right)\right\rangle = \exp\left(\sum\_{m=2} \frac{\lambda\_{m,0}\left(t\right)}{m!} \frac{\hat{\sigma}^m}{\hat{\sigma}q^m}\right) V\left(q\right). \tag{11}$$

Thus, when the anharmonicity of the potential is remarkable, it is expected that the higherorder cumulants play important role in their dynamics. Indeed, for the harmonic oscillator case, only the second-order cumulant appears as

$$
\left\langle V\left(\hat{q}\left(t\right)\right) \right\rangle = \frac{m\alpha^2}{2} \left( q\left(t\right)^2 + \lambda\_{2,0}\left(t\right) \right) \tag{12}
$$

and the other higher-order terms do not.

Up to the second-order, Heisenberg' EOMs for cumulants are given by

$$\begin{aligned} \dot{q}\left(t\right) &= \frac{p\left(t\right)}{m} \\ \dot{p}\left(t\right) &= -\dot{\mathcal{V}}^{(1,0)}\left(q\left(t\right), \mathcal{\lambda}\_{2,0}\left(t\right)\right) \\ \dot{\mathcal{\lambda}}\_{2,0}\left(t\right) &= \frac{2\mathcal{\lambda}\_{1,1}\left(t\right)}{m} \\ \dot{\mathcal{\lambda}}\_{1,1}\left(t\right) &= \frac{\mathcal{\lambda}\_{0,2}\left(t\right)}{m} - \mathcal{\lambda}\_{2,0}\left(t\right)\tilde{\mathcal{V}}^{(2,0)}\left(q\left(t\right), \mathcal{\lambda}\_{2,0}\left(t\right)\right) \\ \dot{\mathcal{\lambda}}\_{0,2}\left(t\right) &= -2\mathcal{\lambda}\_{1,1}\left(t\right)\tilde{\mathcal{V}}^{(2,0)}\left(q\left(t\right), \mathcal{\lambda}\_{2,0}\left(t\right)\right) \end{aligned} \tag{13}$$

where *<sup>V</sup>*˜ is second-order "quantal" potential defined as

term with the potential being a series of *q*. For example, Prezhdo and co-workers derived

by truncating the potential term up to fourth-order power series. The subscript s represent a

ing from previous works, this formalism is essentially the same as Gaussian wave packet method. Prezhdo also proposed a correction to the higher-order moments [8]. Nevertheless their formalism could not be applied general potential without any approximation such as

Recently Shigeta and co-workers derived a general expression for the expectation value of an arbitrary operator by means of cumulants rather than moments [9-19]. For one-dimen‐

sional case, the expectation value of a differential arbitrary operator, *A*<sup>s</sup>

( ( ) ( )) ( )

0

where we introduced the general expression for the cumulant *λm*,*n*(*t*)≡ (*δq*

*m lm*

*A qt pt Aqp lm l q p* l - - £ £ æ ö ¶ <sup>=</sup> ç ÷ - ¶ ¶ è ø

ˆ ˆ, exp , , ! !

which the subscripts mean *m*-th order and *n*-th order with respect to the coordinate and mo‐ mentum, respectively [20-22]. Using the expression, the expectation value of the potential is

> ( ( )) ( ) ( ) ,0 2 <sup>ˆ</sup> exp . !

( ( )) ( ( ) ( )) <sup>2</sup> <sup>2</sup> 2,0 <sup>ˆ</sup> , <sup>2</sup> *<sup>m</sup> V qt qt t* w

= +

Up to the second-order, Heisenberg' EOMs for cumulants are given by

l

Thus, when the anharmonicity of the potential is remarkable, it is expected that the higherorder cumulants play important role in their dynamics. Indeed, for the harmonic oscillator

*V qt V q*

= æ ö ¶ <sup>=</sup> ç ÷ ¶ è ø

*<sup>m</sup> <sup>m</sup> <sup>m</sup> <sup>m</sup> t*

*m q*

l

the symmetric sum of power series of *q* and *p* is derived as

^2(*t*) , and (*<sup>q</sup>*

^(*t*)*<sup>p</sup>* ^(*t*)) <sup>s</sup> <sup>=</sup> <sup>1</sup> <sup>2</sup> (*q* ^(*t*)*<sup>p</sup>*

( ) ( ) ,

*l ml*

å å (10)

å (11)

(12)

*<sup>m</sup> lm l*

*t*

^(*t*)*<sup>p</sup>* ^(*t*))

<sup>s</sup> and solved the EOMs

^(*t*)*<sup>q</sup>*

^(*t*)) . Judg‐

^), that consists of

^*m*(*t*)*<sup>δ</sup> <sup>p</sup>*

^*n*(*t*))s , in

^(*t*) <sup>+</sup> *<sup>p</sup>*

(*q* ^, *<sup>p</sup>*

^2(*t*) , *<sup>p</sup>*

EOMs for three additional moments of *q*

s

case, only the second-order cumulant appears as

and the other higher-order terms do not.

the truncation.

296 Advances in Quantum Mechanics

symmetric sum of the operator product defined as (*q*

$$\left| \tilde{V} \left( q(t), \lambda\_{2,0} \left( t \right) \right) \right| = \left| V \left( \hat{q}(t) \right) \right|\_2 = \exp \left( \frac{\lambda\_{2,0}}{2} \frac{\hat{\sigma}^2}{\hat{\alpha} q^2} \right) V \left( q \right) \Big|\_{q = \left\{ \hat{q}(t) \right\}}.\tag{14}$$

*<sup>V</sup>*˜(*n*,0) is the *n*-th derivative of *<sup>V</sup>*˜ with respect to *<sup>q</sup>*. It is easily seen that the quantal potential is a finite series with respect to the cumulant by expanding as a Taylor series as

$$\left\langle V\left(\hat{q}\left(t\right)\right)\right\rangle\_2 = V\left(\left\langle \hat{q}\left(t\right) \right\rangle\right) + \frac{\lambda\_{2,0}}{2} V^{(2)}\left(\left\langle \hat{q}\left(t\right) \right\rangle\right) + \frac{\lambda\_{2,0}^2}{8} V^{(4)}\left(\left\langle \hat{q}\left(t\right) \right\rangle\right) + \cdots,\tag{15}$$

It is noteworthy that the first and second terms of above equation corresponds to the first and second terms of Eq. (2-9), on the other hand, the other term are different each other.

Now we here give an expression to the quantal potential that has complicated form like as in Eq. (2-14). By using the famous formula for the Gaussian integral

$$\int\_{-\infty}^{\infty} \exp\left[-\left(ar^2 + br\right)\right] dr = \int\_{-\infty}^{\infty} \exp\left[-a\left(r + \frac{b}{2a}\right)^2 + \frac{b^2}{4a}\right] dr = \sqrt{\frac{\pi}{a}} \exp\left(\frac{b^2}{4a}\right) \tag{16}$$

the exponential operator appearing in Eq. (2-14) is rewritten as,

$$\exp\left(\frac{\lambda\_{2,0}}{2}\frac{\partial^2}{\partial q^2}\right) = \frac{1}{\sqrt{2\pi\lambda\_{2,0}}}\int\_{-\infty}^{v} \exp\left[-\left(\frac{r^2}{2\lambda\_{2,0}} + r\frac{\partial}{\partial q}\right)\right] dr.\tag{17}$$

The first derivative operator term in right hand side of the above equation can act to the po‐ tential with the relationship of *exp r* ∂ <sup>∂</sup>*<sup>q</sup> <sup>f</sup>* (*q*)= *<sup>f</sup>* (*<sup>q</sup>* <sup>+</sup> *<sup>r</sup>*) as

$$\begin{split} \exp\left(\frac{\lambda\_{2,0}}{2}\frac{\hat{\sigma}^2}{\hat{c}q^2}\right)V\left(q\right) &= \int\_{-\infty}^{\infty} \frac{dr}{\sqrt{2\pi\lambda\_{2,0}}} \exp\left[-\frac{r^2}{2\lambda\_{2,0}}\right] V\left(q+r\right) \\ &= \int\_{-\infty}^{\infty} \frac{dr}{\sqrt{2\pi\lambda\_{2,0}}} \exp\left[-\frac{\left(q-r\right)^2}{2\lambda\_{2,0}}\right] V\left(r\right). \end{split} \tag{18}$$

( ) ( )

l

l

0,2

l

*t*

1,1

Now by considering the dimension we define new coordinate and momentum as

*p t*

l

l

l

0,2

Above expression indicates that the energy does not depend on *λ*1,1

( ) ( ) ( ( ))

<sup>1</sup> exp 2 2 <sup>2</sup>

l

Total energy are expressed using the cumulant variables as

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

& & & &

0,2

l

&

l

2 2

servation law is give below.

2

0

ò

=

pl

2

The second-order momentum cumulant *λ*0,2

= +

*t*

( ) ( )

l

1,1 2,0 2,0 .

l

( ) ( )

l

( ) ( ) ( )

2 0,2 <sup>2</sup> . <sup>4</sup> *t pt q t* l

*pt t dr r qt Et H V r*

æ ö <sup>+</sup> - ç ÷ <sup>=</sup> =+ -ç ÷

= +

*qt t*

=

=

( ) ( )

 l

( )

2

l

( )

*m t t*

2,0 2,0 exp . <sup>2</sup> <sup>2</sup> <sup>2</sup>

<sup>2</sup> <sup>2</sup>

pl

energy with respect to time gives the energy conservation law. The proof of the energy con‐

( )

è øè ø

*m t t t*

( )

*dr r qt r qt <sup>t</sup> V r t t t*

*ptpt t dr r qt r qt E t q t V r*

æ ö <sup>+</sup> - - ç ÷ <sup>=</sup> - -ç ÷

pl

2,0 <sup>3</sup> 2,0 2,0 2,0

l

æ öæ ö - - ç ÷ç ÷ + +-

ò

( ( )) ( )

exp . <sup>2</sup> <sup>2</sup> <sup>2</sup>

l

2 2

( ( ))

 l

2,0 2,0 2,0

(*t*) is rewritten using them as

*t*

*t*

*t t*

. <sup>4</sup>

<sup>h</sup> (21)

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Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

<sup>h</sup> (23)

( ) ( )

( ( ))

è ø

 l

( ) ( )

( ) ( )

2

(*t*). Differentiating the

(25)

( ( ))

ò (24)

l

è ø

(22)

299

2 2

2,0 2,0

Therefore it is possible to estimate potential energy term without the truncation of the po‐ tential. However the analytic integration is not always has the closed form and the numeri‐ cal integration does not converge depending on the kind of the potential. For the quantal potential including third and higher-order culumant, it is convenient to use the Fourier inte‐ gral instead of Gaussian integral. Nevertheless this scheme also has problems concerning about the integrability and its convergence.

#### **2.3. Energy conservation law and least uncertainty state**

For the EOMs of Eq. (2-13), there exists first integral that always hold for. Now defining a function,

$$
\boldsymbol{\chi}\boldsymbol{\chi}(t) = \boldsymbol{\uplambda}\_{2,0}(t)\boldsymbol{\uplambda}\_{0,2}(t) - \boldsymbol{\uplambda}\_{1,1}^2(t), \tag{19}
$$

and differentiating it result in

$$\begin{split} \dot{\gamma}(t) &= \dot{\lambda}\_{2,0}(t)\lambda\_{0,2}(t) + \lambda\_{2,0}(t)\dot{\lambda}\_{0,2}(t) - 2\lambda\_{1,1}(t)\dot{\lambda}\_{1,1}(t) \\ &= \frac{2\lambda\_{1,1}(t)}{m}\lambda\_{0,2}(t) - 2\lambda\_{2,0}\dot{\lambda}\_{1,1}(t)V^{(2,0)}(q(t), \lambda\_{2,0}(t)) \\ &- 2\lambda\_{1,1}(t)\left| \frac{\lambda\_{0,2}(t)}{m} - \lambda\_{0,2}(t)V^{(2,0)}(q(t), \lambda\_{2,0}(t)) \right| \\ &= 0. \end{split} \tag{20}$$

Thus, this function is a time-independent constant. It is well-known that the least uncertain‐ ty state fulfills *<sup>γ</sup>* <sup>=</sup> <sup>ℏ</sup><sup>2</sup> <sup>4</sup> . By setting the adequate parameter, one can incorporate the Heisen‐ berg' uncertainty principle and thus least uncertainty relation into EOMs. Using this value, one can delete one cumulant from EOMs, for example

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem http://dx.doi.org/10.5772/53703 299

$$
\lambda\_{0,2}\left(t\right) = \frac{\lambda\_{1,1}^2\left(t\right)}{\lambda\_{2,0}\left(t\right)} + \frac{\hbar^2}{4\lambda\_{2,0}\left(t\right)}.\tag{21}
$$

Now by considering the dimension we define new coordinate and momentum as

$$\begin{aligned} p\_{\mathbb{A}}\left(t\right) &= \frac{\mathbb{A}\_{1,1}\left(t\right)}{\sqrt{\mathbb{A}\_{2,0}\left(t\right)}}\\ q\_{\mathbb{A}}\left(t\right) &= \sqrt{\mathbb{A}\_{2,0}\left(t\right)}. \end{aligned} \tag{22}$$

The second-order momentum cumulant *λ*0,2 (*t*) is rewritten using them as

$$
\mathcal{A}\_{0,2}\left(t\right) = p\_{\mathcal{A}}^2\left(t\right) + \frac{\hbar^2}{4q\_{\mathcal{A}}^2\left(t\right)}.\tag{23}
$$

Total energy are expressed using the cumulant variables as

The first derivative operator term in right hand side of the above equation can act to the po‐

<sup>∂</sup>*<sup>q</sup> <sup>f</sup>* (*q*)= *<sup>f</sup>* (*<sup>q</sup>* <sup>+</sup> *<sup>r</sup>*) as

2 2

æ ö ¶ é ù ç ÷ = -+ ê ú ç ÷ è ø ¶ ê ú ë û

pl

pl

Therefore it is possible to estimate potential energy term without the truncation of the po‐ tential. However the analytic integration is not always has the closed form and the numeri‐ cal integration does not converge depending on the kind of the potential. For the quantal potential including third and higher-order culumant, it is convenient to use the Fourier inte‐ gral instead of Gaussian integral. Nevertheless this scheme also has problems concerning

For the EOMs of Eq. (2-13), there exists first integral that always hold for. Now defining a

 l

( ) ( ) ( ) ( ) <sup>2</sup> 2,0 0,2 1,1

( ) ( ) ( ) ( ) ( ) ( ) ( )

*t t t t t tt*

2,0 0,2 2,0 0,2 1,1 1,1

 l

0,2 2,0 1,1 2,0

ï ï î þ

*t tV qt t*

Thus, this function is a time-independent constant. It is well-known that the least uncertain‐

berg' uncertainty principle and thus least uncertainty relation into EOMs. Using this value,

2 ,

*t tV qt t*

2

 ll

( ) ( ) ( ) ( )

0,2 2,0 1,1 0,2 2,0

ì ü ï ï - - í ý

1,1 2,0

 ll

 l

& && &

=+-

( ) ( ) ( ) ( )

*t*

*m*

2 ,

gll

( ) ( )

*dr r V q Vq r*

é ù - <sup>=</sup> ê ú -

2,0 2,0

2,0 2,0

exp . <sup>2</sup> <sup>2</sup>

*dr q r V r*

l

l

ê ú ë û

( ) ( )

*t tt t* = - , (19)

( ( ) ( ))

l

( ( ) ( ))

 l

<sup>4</sup> . By setting the adequate parameter, one can incorporate the Heisen‐

2

(18)

(20)

∂

exp exp 2 2 <sup>2</sup>

¥ -¥

ò

¥ -¥

ò

tential with the relationship of *exp r*

298 Advances in Quantum Mechanics

2,0

l

about the integrability and its convergence.

2

l

 l

*t*

= -

l

l

ll

 l

*m*

one can delete one cumulant from EOMs, for example

0.

=

and differentiating it result in

ty state fulfills *<sup>γ</sup>* <sup>=</sup> <sup>ℏ</sup><sup>2</sup>

g

function,

**2.3. Energy conservation law and least uncertainty state**

2

*q*

$$E\_2(t) = \left\{ H \right\}\_2 = \frac{p^2\left(t\right) + \lambda\_{0,2}\left(t\right)}{2m} + \int \frac{dr}{\sqrt{2\pi\lambda\_{2,0}\left(t\right)}} \exp\left[-\frac{\left(r - q\left(t\right)\right)^2}{2\lambda\_{2,0}\left(t\right)}\right] V\left(r\right). \tag{24}$$

Above expression indicates that the energy does not depend on *λ*1,1 (*t*). Differentiating the energy with respect to time gives the energy conservation law. The proof of the energy con‐ servation law is give below.

$$\begin{split} \dot{E}\_{2}(t) &= \frac{2\dot{p}(t)p(t) + \dot{\lambda}\_{0,2}(t)}{2m} - \dot{q}(t) \Big[ \frac{dr}{\sqrt{2\pi\dot{\lambda}\_{2,0}(t)}} \frac{\left(r - q(t)\right)}{\lambda\_{2,0}(t)} \exp\left(-\frac{\left(r - q(t)\right)^{2}}{2\lambda\_{2,0}(t)}\right) V(r) \\ &+ \Big[ \frac{dr}{\sqrt{2\pi\dot{\lambda}\_{2,0}^{3}(t)}} \dot{\lambda}\_{2,0}(t) \Big] \left(1 + \frac{\left(r - q(t)\right)^{2}}{2\lambda\_{2,0}(t)}\right) \exp\left(-\frac{\left(r - q(t)\right)^{2}}{2\lambda\_{2,0}(t)}\right) V(r) \\ &= 0 \end{split} \tag{25}$$

By means of the new coordinate and momentum, the total energy is rewritten as

$$E\_2 = \frac{p^2\left(t\right) + p\_\lambda^2\left(t\right)}{2m} + \frac{\hbar^2}{8m q\_\lambda^2\left(t\right)} + \frac{1}{q\_\lambda\left(t\right)} \int \frac{dr}{\sqrt{2\pi}} \exp\left[-\frac{\left(r - q\left(t\right)\right)^2}{2q\_\lambda^2\left(t\right)}\right] V\left(r\right). \tag{26}$$

This equation tells us that the effective potential derived from the kinetic energy term affect the dynamics of *q*(*t*) via dynamics of *qλ*(*t*). A variational principle of *E*2,

$$\frac{\partial E\_2}{\partial p} = \frac{\partial E\_2}{\partial p\_{\lambda}} = \frac{\partial E\_2}{\partial q} = \frac{\partial E\_2}{\partial q\_{\lambda}} = 0,\tag{27}$$

particle at *r* is the expectation value of the density operator, *δ*(*q*

Thus the second-order expression for the density is evaluated as

pl

average of the potential with weight *ρ*<sup>2</sup>

tion on the basis of the present theory as

The second-order expression is given by

p g

joint

r

*r s*

r

distribution.

r

 d

2

r

( ) ( ) ( ( ) ) <sup>2</sup> *r qr* ˆ ˆ lim exp *q r* . b

( ) ( )

*r r q*

pl

2,0 2,0 1 1

( ) ( ) ( ) ( ) ( ) 2

l

<sup>ˆ</sup> exp . <sup>2</sup> <sup>2</sup> *dr r q V q V r dr r V r*

This expression indicates that the expectation value of the potential is related to the mean

In principle, one cannot determine the position and momentum at the same time within the quantum mechanics. In other words, resolution of phase space is no more than the Planck' constant, *h*. In contrast to the quantum mechanics, we can define the joint distribution func‐

joint( ) ( ( ) ( ))<sup>s</sup>

( ) ( ) ( )( ) ( ) 2 2

ll

, exp . <sup>2</sup> <sup>2</sup>

é ù - - - -+ - <sup>=</sup> ê ú -

0,2 1,1 2,0

*rq rqsp sp*

g

ë û

 dd

1 2

2 2 2,0 2,0

æ ö - ç ÷ = -º ç ÷ è ø

æ ö = -- ç ÷

exp . <sup>2</sup> <sup>2</sup>

This density shows that the distribution is a Gaussian centered at *q* with a width depending on the cumulant *λ*2,0. Thus the physical meaning of the second-order cumulant *λ*2,0 results in the width of the distribution. As the integration of this density for the whole space becomes unity, the density is normalized. Therefore the density has the physical meaning of probabil‐ ity. Comparison with Eq. (2-18), the potential energy is rewritten by means of the density as

l

è ø

b

p

b

®¥ = -= - - (29)

2

r

(*r*). The same relationship holds for the momentum

*rs q r p s* , . =- - ˆ ˆ (32)

 l

ò ò (31)

as

^−*r*), with a useful expression

http://dx.doi.org/10.5772/53703

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

(30)

301

(33)

gives stationary state that fulfills the least uncertainty condition as

$$\begin{aligned} \frac{\partial E\_2}{\partial p} &= \frac{p}{m} = 0. \\ \frac{\partial E\_2}{\partial p\_\lambda} &= \frac{p\_\lambda}{m} = 0 \\ \frac{\partial E\_2}{\partial q} &= V\_2^{(1,0)} \left(q, q\_\lambda\right) = 0 \\ \frac{\partial E\_2}{\partial q\_\lambda} &= -\frac{\hbar^2}{4mq\_\lambda^3} + q\_\lambda V\_2^{(2,0)} \left(q, q\_\lambda\right) = 0 \end{aligned} \tag{28}$$

For both momenta, the solutions of the above variational principle are zero. On the other hand, the solutions for the coordinates strongly depend on the shape of the give potential. As an exactly soluble case, we here consider the harmonic oscillator. The variational condi‐ tion gives a set of solutions as (*p*, *<sup>p</sup>λ*, *<sup>q</sup>*, *<sup>q</sup>λ*) =(0, 0, 0, <sup>ℏ</sup> / <sup>2</sup>*mω*). The corresponding energy *<sup>E</sup>*<sup>2</sup> <sup>=</sup> <sup>ℏ</sup>*<sup>ω</sup>* <sup>2</sup> is the same as the exact ground state energy. The cumulant variables estimated from the solutions result in (*λ*2,0, *<sup>λ</sup>*1,1, *<sup>λ</sup>*0,2) =( <sup>ℏ</sup> <sup>2</sup>*m<sup>ω</sup>* , 0, *<sup>m</sup>*ℏ*<sup>ω</sup>* <sup>2</sup> ) being the exact expectation val‐ ues for the ground state. Thus the present scheme with the least uncertainty relation is rea‐ sonable at least for the ground state.

#### **2.4. Distribution function and joint distribution**

In order to visualize the trajectory in this theory, we here introduce distribution function as a function of coordinate and second-order cumulant variables. Now the density finding a particle at *r* is the expectation value of the density operator, *δ*(*q* ^−*r*), with a useful expression as

$$\varphi\left(r\right) = \left\langle \mathcal{S}\left(\hat{q} - r\right) \right\rangle = \lim\_{\beta \to \infty} \sqrt{\frac{\beta}{\pi}} \left\langle \exp\left(-\beta\left(\hat{q} - r\right)^2\right) \right\rangle. \tag{29}$$

Thus the second-order expression for the density is evaluated as

By means of the new coordinate and momentum, the total energy is rewritten as

( ) ( )

ò <sup>h</sup>

2 2 2

l

the dynamics of *q*(*t*) via dynamics of *qλ*(*t*). A variational principle of *E*2,

gives stationary state that fulfills the least uncertainty condition as

1,0 2 2

*<sup>E</sup> V qq*

¶ = = ¶

l

2

*E p p m E p p m*

¶ = = ¶

¶ = = ¶

2

l

*q*

¶

¶

from the solutions result in (*λ*2,0, *<sup>λ</sup>*1,1, *<sup>λ</sup>*0,2) =( <sup>ℏ</sup>

**2.4. Distribution function and joint distribution**

sonable at least for the ground state.

*<sup>E</sup>*<sup>2</sup> <sup>=</sup> <sup>ℏ</sup>*<sup>ω</sup>*

l ( ) ( )

l

*<sup>E</sup> qV qq*

, 0

l

=- + =

For both momenta, the solutions of the above variational principle are zero. On the other hand, the solutions for the coordinates strongly depend on the shape of the give potential. As an exactly soluble case, we here consider the harmonic oscillator. The variational condi‐ tion gives a set of solutions as (*p*, *<sup>p</sup>λ*, *<sup>q</sup>*, *<sup>q</sup>λ*) =(0, 0, 0, <sup>ℏ</sup> / <sup>2</sup>*mω*). The corresponding energy

<sup>2</sup> is the same as the exact ground state energy. The cumulant variables estimated

ues for the ground state. Thus the present scheme with the least uncertainty relation is rea‐

In order to visualize the trajectory in this theory, we here introduce distribution function as a function of coordinate and second-order cumulant variables. Now the density finding a

<sup>2</sup>*m<sup>ω</sup>* , 0, *<sup>m</sup>*ℏ*<sup>ω</sup>*

<sup>2</sup> ) being the exact expectation val‐

l

0.

0

2 2,0 2 3 2

h

4

*q mq*

<sup>2</sup> 2 2 <sup>2</sup>

*pt pt dr r qt <sup>E</sup> V r m mq t q t q t*

æ ö <sup>+</sup> - ç ÷ = ++ -ç ÷

l

1

<sup>2222</sup> 0, *EEEE pp qq* l

exp . <sup>2</sup> 8 2 <sup>2</sup>

This equation tells us that the effective potential derived from the kinetic energy term affect

p

 l

( ) ( )

, 0

 l ( ( ))

 l

è ø

¶¶¶¶ ==== ¶¶ ¶¶ (27)

( ) ( )

(26)

(28)

( ) ( )

300 Advances in Quantum Mechanics

l

$$\rho\_2\left(r\right) = \frac{1}{\sqrt{2\pi\lambda\_{2,0}}} \exp\left(-\frac{1}{2\lambda\_{2,0}}\left(r-q\right)^2\right). \tag{30}$$

This density shows that the distribution is a Gaussian centered at *q* with a width depending on the cumulant *λ*2,0. Thus the physical meaning of the second-order cumulant *λ*2,0 results in the width of the distribution. As the integration of this density for the whole space becomes unity, the density is normalized. Therefore the density has the physical meaning of probabil‐ ity. Comparison with Eq. (2-18), the potential energy is rewritten by means of the density as

$$\left\langle V\left(\hat{\boldsymbol{\eta}}\right)\right\rangle\_2 = \int \frac{d\boldsymbol{r}}{\sqrt{2\pi\lambda\_{2,0}}} \exp\left(-\frac{\left(\boldsymbol{r}-\boldsymbol{q}\right)^2}{2\lambda\_{2,0}}\right) V\left(\boldsymbol{r}\right) \equiv \int d\boldsymbol{r} \,\rho\_2\left(\boldsymbol{r}\right) V\left(\boldsymbol{r}\right). \tag{31}$$

This expression indicates that the expectation value of the potential is related to the mean average of the potential with weight *ρ*<sup>2</sup> (*r*). The same relationship holds for the momentum distribution.

In principle, one cannot determine the position and momentum at the same time within the quantum mechanics. In other words, resolution of phase space is no more than the Planck' constant, *h*. In contrast to the quantum mechanics, we can define the joint distribution func‐ tion on the basis of the present theory as

$$\rho\_{\text{joint}}\left(r,s\right) = \left\langle \left(\mathcal{S}\left(\hat{q}-r\right)\mathcal{S}\left(\hat{p}-s\right)\right)\_{\text{s}}\right\rangle. \tag{32}$$

The second-order expression is given by

$$\rho\_{\rm joint}\left(r,s\right) = \frac{1}{2\pi\sqrt{\gamma}}\exp\left[-\frac{\lambda\_{0,2}\left(r-q\right)^2 - 2\lambda\_{1,1}\left(r-q\right)\left(s-p\right) + \lambda\_{2,0}\left(s-p\right)^2}{2\gamma}\right].\tag{33}$$

In contrast to the energy, the joint distribution depends on all the cumulant variables. In the phase space, this joint distribution has the elliptic shape rotated toward *r-s* axes. This joint distribution corresponds not to a simple coherent state, but to a squeezed-coherent state.

Using the joint distribution, the expectation value of the arbitrary operator is evaluated via

$$
\left\langle A\_s(\hat{\boldsymbol{q}}, \hat{\boldsymbol{p}}) \right\rangle = \int \int A(\boldsymbol{r}\_\prime \boldsymbol{s}) \rho\_{\hat{\boldsymbol{p}} \text{init}}(\boldsymbol{r}\_\prime \boldsymbol{s}) d\boldsymbol{r} d\boldsymbol{s}.\tag{34}
$$

( ) ( ) ( )

*q q p p q p*

ˆ ˆ

ˆ ˆ

ˆ ˆ ,

, , <sup>2</sup>

where *PI* and *QI* are *n*-dimensional momentum and coordinae and **1***<sup>n</sup>* =(1 1 ⋯ 1) is *n*-di‐

<sup>1</sup> , exp , <sup>2</sup> 2 det

where *ξ* is an *n by n* matrix composed of the position cumulant variables. From Heisenberg uncertainty relation and the least uncertainty, the total energy of Eq. (2-39) is rewritten as

Tr , . 2 8


({ } { }) ( )

é ù = - - ++ - + ê ú ë û

, ,

, ,

*I kl I ml I J I J I mk I J IJ*

x x

*Im Ik Im Il*

**2 2**

x x

*W W*

**q q**

({ } { })

= + + -+ åå å **<sup>P</sup> Q Q** <sup>h</sup> % x


( ) ( ) ( ) ( ) <sup>1</sup>

*T*

x

( ) ( )

, ,

z

**q q q q**

*I I JI J*

(40)

x x

({ } { })

x x

( )

*I JI J*

x x (38)

303

(39)

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Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

(41)

(42)

, s , s , s

*I kl Ik Il I kl Ik Il I kl Ik Il*

 dd

 dd

= + -+ å å **P 1** % **Q Q**

mensional identity vector. *V* (*Q*, *ξ*) is the second-order quantal potential given as

*<sup>d</sup> V V*

x

 dd

= = =

x

h

z

the total energy is derived as an extension of Eq. (2-26) by

1

( )

x

2

( )

**1**

*Ik*

*I kl I lk*

 z

*I*

*m*

 z

*I m*

 x

å

*Ik I J IJ*

=- - +

**q q**

2 , ,

*Ik*

*I*

, , , 2

*m*

h

*m I kl*

å

,

x

&

h

z

*I kl*

*Ik*

&

*<sup>p</sup> <sup>q</sup> m*

=

*p W*

% &

z

<sup>+</sup> <sup>=</sup>

2

*n*

2 2 LQ 1 2 2

*N N*

*I I I i I J E V m m*

From Heisenberg EOM, EOMs up to the second-order cumulants are given by

= >

1

*I*

({ } { })

,

x x

( )

*I kl i km I J IJ*

**2**

=- - +

,

*Im Il*

, ,2 , 2

% % &

( )

*W*

%

p

2 2

h

*I I I J E V* = > *m* + ×

*N N I In*

In this sense, this theory is one of variants of the quantum distribution function theory. This joint distribution fulfills the following relations as

$$\begin{cases} \rho\_{\text{joint}}(r,s)ds = \rho(r) \\ \int \rho\_{\text{joint}}(r,s)dr = \rho\_{\text{momentum}}(s) \\ \int \int \rho\_{\text{joint}}(r,s)dr ds = 1. \end{cases} \tag{35}$$

Moreover the coordinate, momentum, and cumulants are derived by means of the joint dis‐ tribution as

$$\begin{split} q &= \iint r \rho\_{\text{joint}}(r,s) dr ds \\ p &= \int s \rho\_{\text{joint}}(r,s) dr ds \\ \mathcal{A}\_{2,0} &= \iint \left(r - q\right)^{2} \rho\_{\text{joint}}(r,s) dr ds \\ \mathcal{A}\_{1,1} &= \iint \left(r - q\right) \left(s - p\right) \rho\_{\text{joint}}(r,s) dr ds \\ \mathcal{A}\_{0,2} &= \iint \left(s - p\right)^{2} \rho\_{\text{joint}}(r,s) dr ds. \end{split} \tag{36}$$

#### **2.5. Extension to multi-dimensional systems**

The Hamiltonian of an *n*-dimensional *N* particle system including a two-body interaction is written by

$$\hat{H} = \sum\_{l=1}^{N} \frac{\hat{\mathbf{P}}\_{l}^{2}}{2m\_{l}} + \sum\_{l>l}^{N} V \left( \left| \hat{\mathbf{Q}}\_{l} - \hat{\mathbf{Q}}\_{l} \right| \right) \tag{37}$$

where *Q* **^** *<sup>I</sup>* =(*q* ^ *<sup>I</sup>* <sup>1</sup>, *q* ^ *<sup>I</sup>* <sup>2</sup>, ⋯, *q* ^ *In*) and *P* **^** *<sup>I</sup>* =(*p* ^ *<sup>I</sup>* <sup>1</sup>, *p* ^ *<sup>I</sup>* 2, ⋯, *p* ^ *In*), and *mI* represent a vector of *I*-th po‐ sition operator, that of momentum operator, and mass, respectively. We here assume that the potential *V* (*r*) is a function of the inter-nuclear distance *r*. Using the definitions of the second-order single-particle cumulants given by

#### Quantal Cumulant Mechanics as Extended Ehrenfest Theorem http://dx.doi.org/10.5772/53703 303

$$\begin{aligned} \xi\_{I,kl} &= \left\{ \left( \delta \hat{\boldsymbol{q}}\_{I\boldsymbol{k}} \delta \hat{\boldsymbol{q}}\_{\boldsymbol{l}\boldsymbol{l}} \right)\_{\mathrm{s}} \right\} \\ \eta\_{I,kl} &= \left\{ \left( \delta \hat{\boldsymbol{p}}\_{I\boldsymbol{k}} \delta \hat{\boldsymbol{p}}\_{\boldsymbol{l}\boldsymbol{l}} \right)\_{\mathrm{s}} \right\} \\ \zeta\_{I,kl} &= \left\{ \left( \delta \hat{\boldsymbol{q}}\_{I\boldsymbol{k}} \delta \hat{\boldsymbol{p}}\_{\boldsymbol{l}\boldsymbol{l}} \right)\_{\mathrm{s}} \right\} \end{aligned} \tag{38}$$

the total energy is derived as an extension of Eq. (2-26) by

In contrast to the energy, the joint distribution depends on all the cumulant variables. In the phase space, this joint distribution has the elliptic shape rotated toward *r-s* axes. This joint distribution corresponds not to a simple coherent state, but to a squeezed-coherent state.

Using the joint distribution, the expectation value of the arbitrary operator is evaluated via

In this sense, this theory is one of variants of the quantum distribution function theory. This

( ) ( )

*r s dr s*

 r

 r

Moreover the coordinate, momentum, and cumulants are derived by means of the joint dis‐

òò (34)

(35)

(36)

( ) ( ) ( ) <sup>s</sup> joint *A q p A r s r s drds* ˆ ˆ, , ,. = r

( ) ( )

= = =

*r s ds r*

joint momentum

, 1.

( ) ( )

, ,

joint joint 2 2,0 joint

r

r

= -

òò òò

= -

1

*In*) and *P* **^** *<sup>I</sup>* =(*p* ^ *<sup>I</sup>* 1, *p* ^

second-order single-particle cumulants given by

*N N I*

*I IJ I H V* = > *m*

òò

*q r r s drds p s r s drds*

1,1 joint 2 0,2 joint

= --

( ) ( )

 r

( ) ( )

 r

( )( ) ( )

*s p r s drds*

*r q r s drds*

,

*r q s p r s drds*

 r

The Hamiltonian of an *n*-dimensional *N* particle system including a two-body interaction is

( ) <sup>2</sup>

*<sup>I</sup>* 2, ⋯, *p* ^

sition operator, that of momentum operator, and mass, respectively. We here assume that the potential *V* (*r*) is a function of the inter-nuclear distance *r*. Using the definitions of the

*I J*

=+ - å å **<sup>P</sup> Q Q** (37)

*In*), and *mI* represent a vector of *I*-th po‐

<sup>ˆ</sup> <sup>ˆ</sup> ˆ ˆ , <sup>2</sup>

,

, .

( )

, ,

*r s drds*

joint

joint

r

= = òò òò

òò

l

l

l

**2.5. Extension to multi-dimensional systems**

r

ò ò

r

joint distribution fulfills the following relations as

tribution as

302 Advances in Quantum Mechanics

written by

where *Q* **^** *<sup>I</sup>* =(*q* ^ *<sup>I</sup>* 1, *q* ^ *<sup>I</sup>* 2, ⋯, *q* ^

$$E\_2 = \sum\_{l=1}^{N} \frac{\mathbf{P}\_l^2 + \boldsymbol{\eta}\_l \cdot \mathbf{1}\_n}{2m\_l} + \sum\_{l>l}^{N} \tilde{V}\_2 \left( \mathbf{Q}\_l - \mathbf{Q}\_{l'} \mathbf{f}\_l + \mathbf{f}\_l \right), \tag{39}$$

where *PI* and *QI* are *n*-dimensional momentum and coordinae and **1***<sup>n</sup>* =(1 1 ⋯ 1) is *n*-di‐ mensional identity vector. *V* (*Q*, *ξ*) is the second-order quantal potential given as

$$\vec{V}\_{2}\left(\mathbf{Q},\mathbf{f}\right) = \int \frac{d\mathbf{r}}{\sqrt{\left(2\pi\right)^{\mathrm{T}}\det|\mathbf{f}|}} \exp\left(-\frac{1}{2}\left(\mathbf{Q}-\mathbf{r}\right)^{\mathrm{T}}\mathbf{f}^{-1}\left(\mathbf{Q}-\mathbf{r}\right)\right) V\left(|\mathbf{r}|\right). \tag{40}$$

where *ξ* is an *n by n* matrix composed of the position cumulant variables. From Heisenberg uncertainty relation and the least uncertainty, the total energy of Eq. (2-39) is rewritten as

$$E\_2^{\rm LQ} = \sum\_{I=1}^{N} \frac{\mathbf{P}\_I^2}{2m\_I} + \sum\_i \frac{\hbar^2}{8m\_I} \operatorname{Tr} \Big( \mathbf{f}\_I^{-1} \Big) + \sum\_{I>I}^{N} \tilde{V}\_2 \Big( \mathbf{Q}\_I - \mathbf{Q}\_{I'} \mathbf{f}\_I^\bullet + \mathbf{f}\_I^\bullet \Big). \tag{41}$$

From Heisenberg EOM, EOMs up to the second-order cumulants are given by

$$\begin{split} \dot{q}\_{lk} &= \frac{p\_{lk}}{m\_{I}} \\ \dot{p}\_{lk} &= -\tilde{\mathbf{V}}\_{2}^{(1\_{lk})} \Big( \big( \mathbf{q}\_{I} - \mathbf{q}\_{I} \big), \big( \mathbf{\dot{\mathbf{g}}}\_{I} + \mathbf{\dot{\mathbf{g}}}\_{I} \big) \Big) \\ \dot{\xi}\_{I,kl} &= \frac{\mathcal{L}\_{I,kl} + \mathcal{L}\_{I,lk}}{m\_{I}} \\ \dot{\eta}\_{I,kl} &= -\sum\_{m} \Big( \mathcal{L}\_{I,ml} \tilde{\mathbf{W}}\_{2}^{(2\_{m,lk})} \big( \big( \mathbf{q}\_{I} - \mathbf{q}\_{I} \big), \big( \mathbf{\dot{\mathbf{g}}}\_{I} + \mathbf{\dot{\mathbf{g}}}\_{I} \big) \Big) + \mathcal{L}\_{I,mk} \tilde{\mathbf{W}}\_{2}^{(2\_{m,l})} \Big( \big( \mathbf{q}\_{I} - \mathbf{q}\_{I} \big), \big( \mathbf{\dot{\mathbf{g}}}\_{I} + \mathbf{\dot{\mathbf{g}}}\_{I} \big) \Big) \Big) \\ \mathcal{L}\_{I,kl} &= \frac{\eta\_{I,kl}}{m\_{I}} - \sum\_{m} \bar{\varepsilon}\_{i,km} \tilde{\mathbf{W}}\_{2}^{(2\_{m,l})} \Big( \big( \mathbf{q}\_{I} - \mathbf{q}\_{I} \big), \big( \mathbf{\dot{\mathbf{g}}}\_{I} + \mathbf{\dot{\mathbf{g}}}\_{I} \big) \Big). \end{split} \tag{42}$$

where *<sup>W</sup>*˜ 2 (**1***Ik* ) ({*Q<sup>I</sup>* <sup>−</sup>*Q<sup>J</sup>* }, {*ξ<sup>I</sup>* <sup>+</sup> *<sup>ξ</sup><sup>J</sup>* }) and *<sup>W</sup>*˜ 2 (**2***Ik* ,*Im*) ({*Q<sup>I</sup>* −*Q<sup>J</sup>* }, {*ξ<sup>I</sup>* + *ξ<sup>J</sup>* }) are the 1st and 2nd deriva‐ tives of the sum of the quantal potentials with respect to the position *qik* and to *qik* and *qim* defined as

$$\begin{split} \tilde{W}\_{2}^{(\mathbf{1}\_{\mathrm{lk}})} \left( \{ \mathbf{Q}\_{I} - \mathbf{Q}\_{I} \} , \{ \mathbf{\pm}\_{I} + \mathbf{\pm}\_{I} \} \right) &= \sum\_{I} \frac{\hat{\sigma} \tilde{V}\_{2} \left( \mathbf{Q}\_{I} - \mathbf{Q}\_{I} , \mathbf{\pm}\_{I} + \mathbf{\pm}\_{I} \right)}{\hat{\alpha} q\_{lk}} \\ \tilde{W}\_{2}^{(\mathbf{2}\_{\mathrm{lk}})} \left( \{ \mathbf{Q}\_{I} - \mathbf{Q}\_{I} \} , \{ \mathbf{\pm}\_{I} + \mathbf{\pm}\_{I} \} \right) &= \sum\_{I} \frac{\hat{\sigma}^{2} \tilde{V}\_{2} \left( \mathbf{Q}\_{I} - \mathbf{Q}\_{I} , \mathbf{\pm}\_{I} + \mathbf{\pm}\_{I} \right)}{\hat{\alpha} q\_{lk} \hat{\alpha} q\_{lm}} . \end{split} \tag{43}$$

mode coupling representation of QFF (*n*MR-QFF) was applied [23], which includes mode

By taking each normal mode as the degree of freedom in the dynamics simulation, the Ham‐

where *VQFF <sup>n</sup>*−mode denotes *n*MR-QFF. In this Hamiltonian we neglected the Watson term, which represents the vibrational-rotational coupling. Mass does not appear in the equations since the QFF normal coordinate is mass weighted. Therefore, the time evolution of varia‐ bles of QCD2 with 1MR-QFF (general expressions are not shown for simplicity) is derived as

mode QFF

( ) ( )

2 ,0 2 ,0

2

2 2

1 ,1 0 ,2 2 ,0 2 ,0

*i i i i i i ii iii i i ii*

*<sup>u</sup> h tQ Q*

=- + + + ê ú ë û

2 . 2

=- ++ + ê ú ë û

*iiii*

For molecules with more than 1 degree of freedom, we applied 3MR-QFF, because it has been shown by various examples that the 3MR-QFF is sufficient to describe fundamental modes as well as more complex overtone modes. The QCD2 and classical simulations were performed numerically with a fourth-order Runge-Kutta integrator. For formaldehyde (CH2O) and formic acid (HCOOH), 3MR-QFF PES was generated at the level of MP2/aug-ccpVTZ [24, 25] using GAMESS [26] and Gaussian03 [27] program packages. In this work, the results obtained by our method are compared with those by vibrational self-consistent field method (VSCF) with full second-order perturbation correction (VPT2), which is based on the

We here present results of the spectral analysis of trajectories obtained from the simulation that can be compared with other theoretical calculations and experimental results. The Four‐ ier transform of any dynamical variables obtained from the trajectories of MD simulations is related to spectral densities. In particular, Fourier transform of velocity autocorrelation func‐ tion gives the density of vibrational states. In addition, the power spectrum of the time ser‐ ies or autocorrelation function of each normal coordinate shows the contribution to frequency peaks of the spectrum obtained from velocity autocorrelation. Here we adopted the latter procedure. The time interval used was 0.1 fs and total time is 1 ps for all MD and

é ù

*<sup>u</sup> h tQ Q*

2 6

l

=- + + + +

*iii iiii i ii i i i i i i i i*

*t u P hQ Q Q Q*

0 ,2 1 ,1 2 ,0

quantum mechanics and accurate enough to treat molecular vibrations.

*i i i i ii iii i i ii*

( )

l

2

2

*iiii*

é ù

( )

l

2

 l

(46)

*<sup>P</sup> HP Q V Q* - = + å (45)

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

http://dx.doi.org/10.5772/53703

305

({ } { }) ({ }) <sup>2</sup>

<sup>ˆ</sup> ˆ ˆ ˆ ˆ , , <sup>2</sup> *i n i i i i*

iltonian for QCD2 with *n*MR-QFF as the potential energy is

2 ,0 1 ,1

=

l

&

&

l

&

*i i*

*Q P*

&

&

=

2

 l

lll

 l *ii ii*

couplings up to *n* modes.

In contrast to the one-dimensional problems, second-order cumulants are represented as matrices. Thus, the total degrees of freedom are 24*N* for 3-dimensional cases. For the latter convenience, we here propose two different approximations. The one is the diagonal ap‐ proximation, where the all off-diagonal elements are neglected, and the spherical approxi‐ mation, where the all diagonal cumulants are the same in addition to the diagonal approximation. In the following, we apply the present methods for several multi-dimension‐ al problems. We hereafter refer our method as QCD2.

#### **3. Applications**

#### **3.1. Application to molecular vibration**

Here we evaluate the vibrational modes from the results obtained from molecular dynamics (MD) simulations. Since the force field based model potentials, which are often used in mo‐ lecular dynamics simulations, are empirical so that they sometimes leads to poor results for molecular vibrations. For quantitative results in any MD study, the accuracy of the PES is the other important requirement as well as the treatment of the nuclear motion. Here we use an efficient representation of the PES derived from *ab initio* electronic structure methods, which is suitable for both molecular vibration and the QCD scheme in principle. In order to include anharmonic effects, multi-dimensional quartic force field (QFF) approximation [23] is applied as

$$V\_{\rm QFF}\left(\left\{\hat{\mathbf{Q}}\_{i}\right\}\right) = V\_0 + \sum\_{i} \frac{h\_{ii}}{2} \hat{\mathbf{Q}}\_i^2 + \sum\_{ijk} \frac{t\_{ijk}}{6} \hat{\mathbf{Q}}\_i \hat{\mathbf{Q}}\_j \hat{\mathbf{Q}}\_k + \sum\_{ijk} \frac{u\_{ijkl}}{24} \hat{\mathbf{Q}}\_i \hat{\mathbf{Q}}\_j \hat{\mathbf{Q}}\_k \hat{\mathbf{Q}}\_{l'} \tag{44}$$

where*V*0, *hii* , *tijk* , and *uijkl* denote the potential energy and its second-, third- and fourth-or‐ der derivatives with respect to a set of normal coordinates {*Q* ^ *i* }, at the equilibrium geometry, respectively. To further reduce the computational cost for multi-dimensional cases, an *n*- mode coupling representation of QFF (*n*MR-QFF) was applied [23], which includes mode couplings up to *n* modes.

where *<sup>W</sup>*˜

defined as

2 (**1***Ik* )

304 Advances in Quantum Mechanics

**3. Applications**

is applied as

where*V*0, *hii*

({*Q<sup>I</sup>* <sup>−</sup>*Q<sup>J</sup>* }, {*ξ<sup>I</sup>* <sup>+</sup> *<sup>ξ</sup><sup>J</sup>* }) and *<sup>W</sup>*˜

**Q Q**

**Q Q**

al problems. We hereafter refer our method as QCD2.

({ }) <sup>2</sup>

der derivatives with respect to a set of normal coordinates {*Q*

( )

**1**

*Ik*

( )

**3.1. Application to molecular vibration**

, *tijk* , and *uijkl*

*Ik Im*

2

**2**

*W*

2

*W*

2 (**2***Ik* ,*Im*)

*I J IJ*

% %


x x


x x

*I J IJ*

% %

tives of the sum of the quantal potentials with respect to the position *qik* and to *qik* and *qim*

({ } { }) ( )

, ,

å

å

In contrast to the one-dimensional problems, second-order cumulants are represented as matrices. Thus, the total degrees of freedom are 24*N* for 3-dimensional cases. For the latter convenience, we here propose two different approximations. The one is the diagonal ap‐ proximation, where the all off-diagonal elements are neglected, and the spherical approxi‐ mation, where the all diagonal cumulants are the same in addition to the diagonal approximation. In the following, we apply the present methods for several multi-dimension‐

Here we evaluate the vibrational modes from the results obtained from molecular dynamics (MD) simulations. Since the force field based model potentials, which are often used in mo‐ lecular dynamics simulations, are empirical so that they sometimes leads to poor results for molecular vibrations. For quantitative results in any MD study, the accuracy of the PES is the other important requirement as well as the treatment of the nuclear motion. Here we use an efficient representation of the PES derived from *ab initio* electronic structure methods, which is suitable for both molecular vibration and the QCD scheme in principle. In order to include anharmonic effects, multi-dimensional quartic force field (QFF) approximation [23]

QFF <sup>0</sup> <sup>ˆ</sup> ˆ ˆˆˆ ˆˆˆ ˆ , 2 6 <sup>24</sup>

*ijk ijkl ii i i ijk ijkl i ijk ijkl*

*V Q V Q QQQ QQQ Q* =+ + + åå å (44)

denote the potential energy and its second-, third- and fourth-or‐

^ *i*

}, at the equilibrium geometry,

*h t u*

respectively. To further reduce the computational cost for multi-dimensional cases, an *n*-

2

*V*

*I JI J*

x x

*q*

¶ ¶ -+

¶ -+

**Q Q**

*q q*

¶ ¶

**Q Q**

*I JI J*

x x

, , .

*V*

*J Ik*

*J Ik Im*

2 2

({ } { }) ( ) ,

({*Q<sup>I</sup>* −*Q<sup>J</sup>* }, {*ξ<sup>I</sup>* + *ξ<sup>J</sup>* }) are the 1st and 2nd deriva‐

(43)

By taking each normal mode as the degree of freedom in the dynamics simulation, the Ham‐ iltonian for QCD2 with *n*MR-QFF as the potential energy is

$$
\hat{H}\left(\left\{\hat{P}\_i\right\}, \left\{\hat{Q}\_i\right\}\right) = \sum\_i \frac{\ddot{P}\_i^2}{2} + V\_{\rm QFF}^{n-\rm mode}\left(\left\{\hat{Q}\_i\right\}\right), \tag{45}
$$

where *VQFF <sup>n</sup>*−mode denotes *n*MR-QFF. In this Hamiltonian we neglected the Watson term, which represents the vibrational-rotational coupling. Mass does not appear in the equations since the QFF normal coordinate is mass weighted. Therefore, the time evolution of varia‐ bles of QCD2 with 1MR-QFF (general expressions are not shown for simplicity) is derived as

$$\begin{aligned} \dot{Q}\_i &= P\_i\\ \dot{P}\_i &= -h\_{ii}Q\_i + \frac{t\_{iii}}{2} \left( Q\_i^2 + \lambda\_{2i, 0i} \right) + \frac{u\_{i\overline{i}\overline{i}i}}{6} Q\_i \left( Q\_i^2 + 2\lambda\_{2i, 0i} \right) \\ \dot{\lambda}\_{2i, 0i} &= 2\lambda\_{1i, 1i} \\ \dot{\lambda}\_{1i, 1i} &= \lambda\_{0i, 2i} - \lambda\_{2i, 0i} \left[ h\_{ii} + t\_{ii\overline{i}} Q\_i + \frac{u\_{i\overline{i}i}}{2} \left( Q\_i^2 + \lambda\_{2i, 0i} \right) \right] \\ \dot{\lambda}\_{0i, 2i} &= -2\lambda\_{1i, 1i} \left[ h\_{ii} + t\_{ii\overline{i}} Q\_i + \frac{u\_{i\overline{i}i}}{2} \left( Q\_i^2 + \lambda\_{2i, 0i} \right) \right] . \end{aligned} \tag{46}$$

For molecules with more than 1 degree of freedom, we applied 3MR-QFF, because it has been shown by various examples that the 3MR-QFF is sufficient to describe fundamental modes as well as more complex overtone modes. The QCD2 and classical simulations were performed numerically with a fourth-order Runge-Kutta integrator. For formaldehyde (CH2O) and formic acid (HCOOH), 3MR-QFF PES was generated at the level of MP2/aug-ccpVTZ [24, 25] using GAMESS [26] and Gaussian03 [27] program packages. In this work, the results obtained by our method are compared with those by vibrational self-consistent field method (VSCF) with full second-order perturbation correction (VPT2), which is based on the quantum mechanics and accurate enough to treat molecular vibrations.

We here present results of the spectral analysis of trajectories obtained from the simulation that can be compared with other theoretical calculations and experimental results. The Four‐ ier transform of any dynamical variables obtained from the trajectories of MD simulations is related to spectral densities. In particular, Fourier transform of velocity autocorrelation func‐ tion gives the density of vibrational states. In addition, the power spectrum of the time ser‐ ies or autocorrelation function of each normal coordinate shows the contribution to frequency peaks of the spectrum obtained from velocity autocorrelation. Here we adopted the latter procedure. The time interval used was 0.1 fs and total time is 1 ps for all MD and QCD simulations. The resolution in the frequency domain is less than 1 cm-1, which is enough accuracy for the analysis of the molecular vibrations of interest. If a longer time tra‐ jectory is obtained, the resolution of the Fourier spectrum becomes fine.

**3.2. Proton transfer reaction in guanine-cytosine base pair**

model Guanine-Cytosine base pair. The model potential is given by

( ) GC

the figure. The corresponding quantal potentials are explicitly given by

GC , , 2 , ,

2 5 25 2 5 2


, , 2, 2 , 2 2! 2 ! 2 !

well({ }) <sup>0</sup> ( ( max ) ( min ))

erf 1 ( ) , <sup>2</sup> *<sup>b</sup> bx*

q

*ijk i lj mk n*

' , !! ! ! ! !

*il jm kn* ++ +

where Greek characters denote the cumulant variables. In order to avoid the particles escap‐

1 , *<sup>i</sup> bi i bi i*

where *qi*max and *qi*min are maximum and minimum range of potential and *V*<sup>0</sup> is height of the

where *b* is an effective width of the approximate Heaviside function and guarantees smooth‐ ness of the potential. Using this approximate Heaviside function, the quantal potential for

é ù =+ - -- ê ú ë û

+

,, 0 0 0 0

== = =

*lmn i j k V xyz H xyz*

( ) ( ) ( ) , ,

ing from the bottoms, we have added the well-like potential is defined as

, ,

*i xyz*

( )

=

well-like potential. An approximate Heaviside function is given by

q*x*

*i lj mk n H h*

( )

with

xhz

*lmn*

*V qV*

DNA base pairs have two and three inter-base hydrogen bonds for Adenine-Thymine and Guanine-Cytosine pairs, respectively. Proton transfer reactions among based were theoreti‐ cally investigated by quantum chemical methods and further quantum mechanical analyses for decases [28-33]. In order to investigate dynamical stability of proton-transferred struc‐ tures of the model system consisting DNA bases, we here perform QCD2 simulations of a

, ,

where parameters in the model potential are given by Villani's paper [30, 31], which is fifthorder polynomials with respect to the coordinates for GC pairs and determined by the first principle calculations (B3LYP/cc-pVDZ). The reaction coordinates *x*, *y*, and *z* are shown in

,,,,, , 22 2

æ öæ ö æ ö <sup>=</sup> ç ÷ç ÷ ç ÷ è øè ø è ø % å ååå (48)

*V xyz h xyz* <sup>=</sup> å (47)

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

http://dx.doi.org/10.5772/53703

307

*lmn lmn lmn i k j ijk*

++ + <sup>=</sup> (49)

 q*qq qq*

Õ (50)

= (51)

xh

 z

, , ,, ' , *i k <sup>j</sup> ijk*

*ijk*

Since each normal mode is taken as the degree of freedom explicitly in the present dynamics simulation, the interpretation and analysis of the results can directly be related with each normal mode. The results are shown in Table 1. The table indicates that the harmonic and QFF approximation of the PES results in a large deviation between each other. Therefore, anharmonicity of the potential must be considered to perform reliable simulations. The table shows that for the analysis of fundamental frequencies, the QCD2 has higher accuracy than the classical results, which can be compared with the VPT2 results in all cases. In spite of the high accuracy, the computational cost of the QCD2 remains low even when applied to larger systems. For HCOOH molecule, the QFF is so anharmonic that the classical simulation does not give clear vibrational frequencies due to the chaotic behavior of the power spectrum. The QCD may suppress the chaotic motion as seen in the full quantum mechanics.


**Table 1.**

#### **3.2. Proton transfer reaction in guanine-cytosine base pair**

DNA base pairs have two and three inter-base hydrogen bonds for Adenine-Thymine and Guanine-Cytosine pairs, respectively. Proton transfer reactions among based were theoreti‐ cally investigated by quantum chemical methods and further quantum mechanical analyses for decases [28-33]. In order to investigate dynamical stability of proton-transferred struc‐ tures of the model system consisting DNA bases, we here perform QCD2 simulations of a model Guanine-Cytosine base pair. The model potential is given by

$$V^{\rm GC}\left(\mathbf{x},y,z\right) = \sum\_{i,j,k} h^\*\_{i,j,k} \mathbf{x}^i y^j z^k,\tag{47}$$

where parameters in the model potential are given by Villani's paper [30, 31], which is fifthorder polynomials with respect to the coordinates for GC pairs and determined by the first principle calculations (B3LYP/cc-pVDZ). The reaction coordinates *x*, *y*, and *z* are shown in the figure. The corresponding quantal potentials are explicitly given by

$$\tilde{V}\_2^{\rm CC}\left(\mathbf{x}, y, z, \boldsymbol{\xi}, \boldsymbol{\eta}, \boldsymbol{\zeta}\right) = \sum\_{l, m, n = 0}^{2} \sum\_{i = 0}^{5 - 2l \cdot 5 - 2m} \sum\_{j = 0}^{5 - 2n} \sum\_{k = 0}^{5 - 2n} H\_{i, j, k}^{l, m, n} \mathbf{x}^{i} y^{j} \boldsymbol{z}^{k} \left(\frac{\boldsymbol{\xi}}{2}\right) \left(\frac{\boldsymbol{\eta}}{2}\right)^{n} \left(\frac{\boldsymbol{\zeta}}{2}\right)^{n},\tag{48}$$

with

QCD simulations. The resolution in the frequency domain is less than 1 cm-1, which is enough accuracy for the analysis of the molecular vibrations of interest. If a longer time tra‐

Since each normal mode is taken as the degree of freedom explicitly in the present dynamics simulation, the interpretation and analysis of the results can directly be related with each normal mode. The results are shown in Table 1. The table indicates that the harmonic and QFF approximation of the PES results in a large deviation between each other. Therefore, anharmonicity of the potential must be considered to perform reliable simulations. The table shows that for the analysis of fundamental frequencies, the QCD2 has higher accuracy than the classical results, which can be compared with the VPT2 results in all cases. In spite of the high accuracy, the computational cost of the QCD2 remains low even when applied to larger systems. For HCOOH molecule, the QFF is so anharmonic that the classical simulation does not give clear vibrational frequencies due to the chaotic behavior of the power spectrum.

The QCD may suppress the chaotic motion as seen in the full quantum mechanics.

H2CO

306 Advances in Quantum Mechanics

HCOOH

**Table 1.**

ν8 /ν<sup>9</sup> 1058 /676

**Mode NMA MD QCD VPT2 Exp.**

ν<sup>1</sup> 3040 2901 2843 2866 2843

ν<sup>2</sup> 2997 2868 2838 2849 2782

ν<sup>3</sup> 1766 1764 1723 1734 1746

ν<sup>4</sup> 1548 1504 1509 1515 1500

ν<sup>5</sup> 1268 1247 1250 1251 1250

ν<sup>6</sup> 1202 1166 N/A 1189 1167

ν<sup>1</sup> 3739 N/A 3527 3554 3570

ν<sup>2</sup> 3126 N/A 2980 2989 2943

ν<sup>3</sup> 1794 N/A 1761 1761 1770

ν<sup>4</sup> 1409 N/A 1377 1385 1387

ν<sup>5</sup> 1302 N/A 1270 1231 1229

ν<sup>6</sup> 1130 N/A 1120 1097 1105

ν<sup>7</sup> 626 N/A 631 620 625

N/A N/A

1036 /642

1033 /638

jectory is obtained, the resolution of the Fourier spectrum becomes fine.

$$H\_{i,j,k}^{l,m,n} = \frac{\left(i+2l\right)!}{i!l!} \frac{\left(j+2m\right)!}{j!m!} \frac{\left(k+2n\right)!}{k!n!} h'\_{i+2l,j+2m,k+2n} \tag{49}$$

where Greek characters denote the cumulant variables. In order to avoid the particles escap‐ ing from the bottoms, we have added the well-like potential is defined as

$$V\_{\text{well}}\left(\left\{q\_{i}\right\}\right) = V\_{0}\left[1 + \prod\_{i=\text{x},y,z} \left(\theta\_{b}\left(q\_{i} - q\_{i\text{max}}\right) - \theta\_{b}\left(q\_{i} - q\_{i\text{min}}\right)\right)\right] \tag{50}$$

where *qi*max and *qi*min are maximum and minimum range of potential and *V*<sup>0</sup> is height of the well-like potential. An approximate Heaviside function is given by

$$\theta\_b(\mathbf{x}) = \frac{\text{erf}\left(\sqrt{b}\mathbf{x}\right) + 1}{2},\tag{51}$$

where *b* is an effective width of the approximate Heaviside function and guarantees smooth‐ ness of the potential. Using this approximate Heaviside function, the quantal potential for the well-like potential is analytically derived. When *V*0 is appropriately large, the particles stay around minima during dynamics simulations. We set *b* =100, *qi*max =2.0(Å), *qi*min =0.4(Å), and *V*<sup>0</sup> =0.05 (a.u.). Both *qi*max and *qi*min are in a reasonable range for the coordinate of the proton, because the distances between heavy elements (O and N) of the DNA bases are ap‐ proximately 2.7~3.0 (Å) and roughly speaking the bond length of OH and NH are almost 1.0 ~ 1.1 (Å). The ordinary PES analysis gives both global and metastable structures for the GC pair. The former structure is the original Watson-Crick type and the latter is double protontransferred one as easily found in (a) and (c). No other proton-transferred structure is found on the PES.

In the actual calculations, the time interval used was 0.1fs, total time is 2ps. The initial condi‐ tions of the variables can be determined by the least quantal energy principle. In figures 2 we have depicted phase space (*x/px, y/py, z/pz*) structures of a trajectory obtained by the QCD simulations. For cases (a) and (b), the dynamical feature of the closed orbits is the same except for its amplitudes. The phase space of the *x/px* is compact, on the other hand, that of *z/pz* is loose in comparison with that of *y/py*. The explicit isotope effects on the phase space structure are found in the cases of (c) and (d). In Fig. 2-(c), the nuclei initially located at the metastable structure go out from the basin and strongly vibrate around the global minimum due to tunneling. On the other hand, the deuterated isotopomer remains around the meta‐ stable structure. It is concluded that the metastable structure of the protonated isotopomer is quantum mechanically unstable, though it is classically stable based on the PES analysis. Therefore, it is important to take the quantum effects into isotope effects on the metastable structure with a small energy gap.

**Figure 2.** QCD phase space structures of the GC pair, where *q*=*x*, *y*, or *z* and *p*=*px*, *py*,

around the metastable structure for the protonated and deuterated cases.

**3.3. Quantal structural transition of finite clusters**

Morse potential has following form:

Gaussian fit for the potential as

initially located around the global minimum for the protonated and deuterated cases. (c) and (d) are initially located

Melting behavior of the finite quantum clusters were extensively investigated by many re‐ searchers using different kind of methodologies [34-39]. We here investigate the melting be‐ havior of *n* particle Morse clusters (abbreviated as Mn) by means of the QCD2 method. The

where *D*e, *R*e and *ρ* are parameters for a depth, a position of minimum, and a curvature of potential. In order to evaluate the quantal potential for the Morse potential, we adopt a

> ( ) <sup>2</sup> 1

=

*G i i V r ce*-

, *<sup>G</sup> i <sup>N</sup> <sup>r</sup>*

a

 r( )) 2exp( (*r R* )) , ë û (52)

<sup>=</sup> å (53)

*V r D rR <sup>M</sup>* ( ) = - - - -- ee e é ù exp 2 ( r

or *pz*, respectively. (a) and (b) are

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309

**Figure 1.** A Model for multiple proton transfer reactions in GC pairs. *x*, *y*, and *z* are reaction coordinates of the proton transfer reactions

**Figure 2.** QCD phase space structures of the GC pair, where *q*=*x*, *y*, or *z* and *p*=*px*, *py*, or *pz*, respectively. (a) and (b) are initially located around the global minimum for the protonated and deuterated cases. (c) and (d) are initially located around the metastable structure for the protonated and deuterated cases.

#### **3.3. Quantal structural transition of finite clusters**

the well-like potential is analytically derived. When *V*0 is appropriately large, the particles stay around minima during dynamics simulations. We set *b* =100, *qi*max =2.0(Å), *qi*min =0.4(Å), and *V*<sup>0</sup> =0.05 (a.u.). Both *qi*max and *qi*min are in a reasonable range for the coordinate of the proton, because the distances between heavy elements (O and N) of the DNA bases are ap‐ proximately 2.7~3.0 (Å) and roughly speaking the bond length of OH and NH are almost 1.0 ~ 1.1 (Å). The ordinary PES analysis gives both global and metastable structures for the GC pair. The former structure is the original Watson-Crick type and the latter is double protontransferred one as easily found in (a) and (c). No other proton-transferred structure is found

In the actual calculations, the time interval used was 0.1fs, total time is 2ps. The initial condi‐ tions of the variables can be determined by the least quantal energy principle. In figures 2 we have depicted phase space (*x/px, y/py, z/pz*) structures of a trajectory obtained by the QCD simulations. For cases (a) and (b), the dynamical feature of the closed orbits is the same except for its amplitudes. The phase space of the *x/px* is compact, on the other hand, that of *z/pz* is loose in comparison with that of *y/py*. The explicit isotope effects on the phase space structure are found in the cases of (c) and (d). In Fig. 2-(c), the nuclei initially located at the metastable structure go out from the basin and strongly vibrate around the global minimum due to tunneling. On the other hand, the deuterated isotopomer remains around the meta‐ stable structure. It is concluded that the metastable structure of the protonated isotopomer is quantum mechanically unstable, though it is classically stable based on the PES analysis. Therefore, it is important to take the quantum effects into isotope effects on the metastable

**Figure 1.** A Model for multiple proton transfer reactions in GC pairs. *x*, *y*, and *z* are reaction coordinates of the proton

on the PES.

308 Advances in Quantum Mechanics

transfer reactions

structure with a small energy gap.

Melting behavior of the finite quantum clusters were extensively investigated by many re‐ searchers using different kind of methodologies [34-39]. We here investigate the melting be‐ havior of *n* particle Morse clusters (abbreviated as Mn) by means of the QCD2 method. The Morse potential has following form:

$$V\_M(r) = D\_\mathbf{e} \left[ \exp\left(-2\rho\left(r - R\_\mathbf{e}\right)\right) - 2\exp\left(-\rho\left(r - R\_\mathbf{e}\right)\right) \right],\tag{52}$$

where *D*e, *R*e and *ρ* are parameters for a depth, a position of minimum, and a curvature of potential. In order to evaluate the quantal potential for the Morse potential, we adopt a Gaussian fit for the potential as

$$V\_G\left(r\right) = \sum\_{i=1}^{N\_G} c\_i e^{-\alpha\_i r^2} \,\,\,\,\,\,\,\,\tag{53}$$

which has an analytic form of the quantal potential and coefficients {*ci* } are obtained by a least square fit for a set of even-tempered exponent with upper and lower bounds (*α*upper=10<sup>6</sup> and *α*lower=10−<sup>3</sup> ). By choosing the number of Gaussians, *NG*, the set of the coeffi‐ cients is explicitly determined and we here set *NG*=41.

The Lindemann indexes obtained by CD and QCD are illustrated in Fig. 1. In the figure, there exist three different regions. Until a freezing point, the Lindemann index gradually in‐ creases as the increase of the additional kinetic energy. In this region, the structural transi‐ tion does not actually occur and the cluster remains stiff. This phase is called "solid-like phase". On the other hand, above a melting point, the structural transition often occurs and the cluster is soft. This phase is called "liquid-like phase". Between two phases, the Linde‐ mann index rapidly increases. This phase is referred as "coexistence phase", which is not al‐ lowed for the bulk systems and peculiar to the finite systems. For both the solid- and liquidlike phases, the Lindeman index does not deviate too much. However that of the coexistence phase fluctuates due to a choice of the initial condition. In comparison with CD and QCD results, the transition temperatures of QCD are lower than those of the CD reflecting the quantum effects. The freezing and melting temperatures are about 0.35 and 0.42 for QCD and about 0.41 and 0.60 for CD, respectively. Since the zero-point vibrational energy is in‐ cluded in QCD, the energy barrier between the basin and transition state become lower so that the less temperature is needed to overcome the barrier. This is so-called quantum soft‐ ening as indicated by Doll and coworker for the Neon case by means of the path-integral approach. Our real-time dynamics well reproduce their tendency for this static property. On

*rij* = |*q<sup>i</sup>* −*q <sup>j</sup>* | is slight different, whereas the transition temperature is the same. In the sol‐ id-like phase the Lindemann index obtained by the classical dynamics is equivalent to that of the classical contribution from QCD approach. On the other hand, in the liquid-like phase the Lindemann index is equivalent to that by the classical dynamics. This fact originates from the fact that the high temperature limit of the quantum results coincides with that of the classical one. It is stressed here that the cumulant variables, which contributes to not on‐

^

**Energy Distance**

**Spherical Diagonal Original Classical Original Classical**

**M3** -2.52135 -2.55978 -2.56510 -3.00000 1.07198 1 **M4** -5.20344 -5.22683 -5.24237 -6.0000 1.07811 1

**Table 2.** Nearest and next nearest distances and energy for global minimum obtained by cumulant and classical

*ij* QCD2 = |*q*

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311

0.99212 1.15085

1.00473 1.37518

0.88288 1.01263

**^** *<sup>i</sup>* −*q*

**^** *<sup>j</sup>* <sup>|</sup> QCD2 and

0.91725 1.08364

0.93581 1.32343

0.81221 0.948237

the other hand, behavior of Lidemann indexes from *r*

ly to the quantum delocalization but also to the thermal fluctuation.

**M5** -8.71392 -8.74464 -8.76049 -9.85233

**M6** -13.1836 -13.2141 -13.2141 -14.7182

**M7** -18.4016 -18.4404 -18.4415 -20.3282

dynamics for 3-dimensional M*n* clusters (*n*=3-7).

We here evaluate optimized structures of Mn clusters (n=3-7) for *De* =1, *R*e=1, and *α* =1. The classical global minimum structures of M3, M4, M5, M6, and M7 structure have C3v, Td, D3h, Oh, and D5h symmetry respectively. Table 2 lists energy for each method. We found that the diagonal approximation causes the artificial symmetry breaking and the spherical approxi‐ mation gives less accurate results. Original approximation gives the most accurate and cor‐ rectly symmetric global minimum structures. The diagonal approximation gives the same results by the original one for M6 due to the same reason denoted before. The error of both the diagonal and spherical approximation decreases with the increase of the number of the particles. It is expected that both approximations work well for many particle systems in‐ stead of the original one. In particular the error of the diagonal approximation is 0.006 % for the M7 cluster. This fact tells us that the diagonal approximation is reliable for the M7 cluster at least the stable structure.

For the analyses on quantum melting behavior, the parameters of the Morse potential are chosen as*De* =1, *R*e=3, and *α* =1. Table 2 also lists nearest and next nearest distances obtained by the original and classical ones. As found in this Table, all the distances elongate with re‐ spect to classical ones. For example, the distances of M3 and M4 elongate by 7.2% and 7.8%, respectively. It is notable that these distances does not equally elongate. The ratio between the original and classical ones is different for the nearest and next nearest distances, i.e. 8.16% and 6.20% for M5, 7.36% and 3.91% for M6, and 8.70% and 6.79% for M7, respectively. In future works, we investigate influence of these behaviors on the structural transition (de‐ formation) of the quantum Morse clusters in detail.

In order to measure the melting behavior of the finite clusters, we here use the Lindemann index defined as

$$
\sigma = \frac{2}{N\left(N-1\right)} \sum\_{ij} \frac{\sqrt{\left - \left^2}}{\left},\tag{54}
$$

where *rij* is a long time-averaged distance between *i* and *j*-th particles. In the present ap‐ proach, there exist two possible choices of the average. One is the quantum mechanical aver‐ age within the second-order QCD approach, *r* ^ *ij* QCD2 = |*q* **^** *<sup>i</sup>* −*q* **^** *<sup>j</sup>* <sup>|</sup> QCD2, which include information of both the classical position and the second-order position cumulant simulta‐ neously, and the other is the average evaluated by means of the classical positions appear‐ ing in the QCD approach, *rij* = |*q<sup>i</sup>* −*q <sup>j</sup>* | . We perform real-time dynamics simulation to obtain the Lindemann index for both systems, where we adopt *m*=100.

The Lindemann indexes obtained by CD and QCD are illustrated in Fig. 1. In the figure, there exist three different regions. Until a freezing point, the Lindemann index gradually in‐ creases as the increase of the additional kinetic energy. In this region, the structural transi‐ tion does not actually occur and the cluster remains stiff. This phase is called "solid-like phase". On the other hand, above a melting point, the structural transition often occurs and the cluster is soft. This phase is called "liquid-like phase". Between two phases, the Linde‐ mann index rapidly increases. This phase is referred as "coexistence phase", which is not al‐ lowed for the bulk systems and peculiar to the finite systems. For both the solid- and liquidlike phases, the Lindeman index does not deviate too much. However that of the coexistence phase fluctuates due to a choice of the initial condition. In comparison with CD and QCD results, the transition temperatures of QCD are lower than those of the CD reflecting the quantum effects. The freezing and melting temperatures are about 0.35 and 0.42 for QCD and about 0.41 and 0.60 for CD, respectively. Since the zero-point vibrational energy is in‐ cluded in QCD, the energy barrier between the basin and transition state become lower so that the less temperature is needed to overcome the barrier. This is so-called quantum soft‐ ening as indicated by Doll and coworker for the Neon case by means of the path-integral approach. Our real-time dynamics well reproduce their tendency for this static property. On the other hand, behavior of Lidemann indexes from *r* ^ *ij* QCD2 = |*q* **^** *<sup>i</sup>* −*q* **^** *<sup>j</sup>* <sup>|</sup> QCD2 and *rij* = |*q<sup>i</sup>* −*q <sup>j</sup>* | is slight different, whereas the transition temperature is the same. In the sol‐ id-like phase the Lindemann index obtained by the classical dynamics is equivalent to that of the classical contribution from QCD approach. On the other hand, in the liquid-like phase the Lindemann index is equivalent to that by the classical dynamics. This fact originates from the fact that the high temperature limit of the quantum results coincides with that of the classical one. It is stressed here that the cumulant variables, which contributes to not on‐ ly to the quantum delocalization but also to the thermal fluctuation.

which has an analytic form of the quantal potential and coefficients {*ci*

(*α*upper=10<sup>6</sup>

310 Advances in Quantum Mechanics

and *α*lower=10−<sup>3</sup>

at least the stable structure.

index defined as

cients is explicitly determined and we here set *NG*=41.

formation) of the quantum Morse clusters in detail.

s

age within the second-order QCD approach, *r*

least square fit for a set of even-tempered exponent with upper and lower bounds

We here evaluate optimized structures of Mn clusters (n=3-7) for *De* =1, *R*e=1, and *α* =1. The classical global minimum structures of M3, M4, M5, M6, and M7 structure have C3v, Td, D3h, Oh, and D5h symmetry respectively. Table 2 lists energy for each method. We found that the diagonal approximation causes the artificial symmetry breaking and the spherical approxi‐ mation gives less accurate results. Original approximation gives the most accurate and cor‐ rectly symmetric global minimum structures. The diagonal approximation gives the same results by the original one for M6 due to the same reason denoted before. The error of both the diagonal and spherical approximation decreases with the increase of the number of the particles. It is expected that both approximations work well for many particle systems in‐ stead of the original one. In particular the error of the diagonal approximation is 0.006 % for the M7 cluster. This fact tells us that the diagonal approximation is reliable for the M7 cluster

For the analyses on quantum melting behavior, the parameters of the Morse potential are chosen as*De* =1, *R*e=3, and *α* =1. Table 2 also lists nearest and next nearest distances obtained by the original and classical ones. As found in this Table, all the distances elongate with re‐ spect to classical ones. For example, the distances of M3 and M4 elongate by 7.2% and 7.8%, respectively. It is notable that these distances does not equally elongate. The ratio between the original and classical ones is different for the nearest and next nearest distances, i.e. 8.16% and 6.20% for M5, 7.36% and 3.91% for M6, and 8.70% and 6.79% for M7, respectively. In future works, we investigate influence of these behaviors on the structural transition (de‐

In order to measure the melting behavior of the finite clusters, we here use the Lindemann

<sup>2</sup> , <sup>1</sup> *ij ij*

*ij ij*

where *rij* is a long time-averaged distance between *i* and *j*-th particles. In the present ap‐ proach, there exist two possible choices of the average. One is the quantum mechanical aver‐

information of both the classical position and the second-order position cumulant simulta‐ neously, and the other is the average evaluated by means of the classical positions appear‐ ing in the QCD approach, *rij* = |*q<sup>i</sup>* −*q <sup>j</sup>* | . We perform real-time dynamics simulation to

<sup>2</sup> <sup>2</sup>

^

*ij* QCD2 = |*q*

<sup>=</sup> - å (54)

**^** *<sup>i</sup>* −*q*

**^** *<sup>j</sup>* <sup>|</sup> QCD2, which include

*r r*


( )

obtain the Lindemann index for both systems, where we adopt *m*=100.

*N N r*

). By choosing the number of Gaussians, *NG*, the set of the coeffi‐

} are obtained by a


**Table 2.** Nearest and next nearest distances and energy for global minimum obtained by cumulant and classical dynamics for 3-dimensional M*n* clusters (*n*=3-7).

As numerical examples, we performed four applications to the simple systems. The first is the application to molecular vibrations. At first we showed that the normal mode analysis is extended to the effective potential appeared in the QCD. We illustrated that the anharmonic contribution is taken into account through mixing between the ordinary and the extended coordinates. The QCD simulations for the *ab initio* derived quartic force field are performed. The vibrational frequencies obtained from its power spectrum are good agreement with

Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

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313

The second is the proton transfer reactions in model DNA base pairs. We numerically showed the geometric isotope effects on the stability of the proton-transferred structures of the DNA base pairs as a function of the mass. We performed QCD simulations in order to investigate dynamical stability of the proton-transferred GC pair. The results showed that the proton-transferred structure of the protonated isotopomer is dynamically unstable and that of deuterated isotopomer remains stable. In former case, dynamically induced transi‐ tion from the metastable to global minimum occurs. It is relevant to include dynamical ef‐

The last application is structural transition of finite quantum Morse clusters. We first com‐ pared the energy of the stable structures of the classical M*n* cluster with those of quantum counterpart and found that the quantum effects due to zero point vibration is remarkable for small system and suppressed for larger. Then we performed the real-time dynamics to evaluate the Lindemann index to characterize the dynamical effects on the melting for M7 cluster. In between solid-like and liquid-like phases (so-called coexistence phase), structural

This study is supported by a Grant-in-Aid for Young Scientists (A) (No. 22685003) from Ja‐ pan Society for the Promotion of the Science (JSPS) and also by a CREST program from Ja‐

1 Department of Materials Engineering Science, Graduate School of Engineering Sciences,

2 Japan Science and Technology Agency, Kawaguchi Center Building, Honcho, Kawaguchi-

Osaka University, Machikaneyama-cho, Toyonaka, Osaka, Japan

those obtained by accurate methods such as VPT2 and VCI.

fects to treat quantum isotope effects on the proton transfer reactions.

changes of the cluster occur intermittently.

pan Science and Technology Agency (JST).

**Acknowledgements**

**Author details**

Yasuteru Shigeta1,2

shi, Saitama, Japan

**Figure 3.** Static Lindemann indexes evaluated from QCD quantum distance (red), QCD classical distance (blue), and CD classical distance (green).

#### **4. Summary**

As an extension to the mechanics concerning about Ehrenfest theorem, we formulated a quantal cumulant mechanics (QCM) and corresponding dynamic method (QCD). The key point is the use of a position shift operator acting on the potential operator and introducing the cumulant variables to evaluate it, so that one need not truncate the potential, and it does not require separating into quantum and classical parts. In particular, we derived the cou‐ pled equation of motion (EOM) for the position, momentum, and second-order cumulants of the product of the momentum and position fluctuation operators. The EOM consists of variables and a quantal potential and its derivatives, where the quantal potential is ex‐ pressed as an exponential function of the differential operator acting on the given potential. We defined density and joint density evaluated from the cumulant expansion scheme. It is clearly found that the present second-order approach gives a Gaussian density distribution spanned both on position and momentum space. Since the density is normalized, the joint density is considered exactly as probability distribution. We also indicated the relation be‐ tween the joint density and cumulant variables as expectation values calculated from the distribution. We extended the QCD for the one-dimensional system in to treat the multi-di‐ mensional systems. We derived the EOMs with the 24*N* dimensional phase space.

As numerical examples, we performed four applications to the simple systems. The first is the application to molecular vibrations. At first we showed that the normal mode analysis is extended to the effective potential appeared in the QCD. We illustrated that the anharmonic contribution is taken into account through mixing between the ordinary and the extended coordinates. The QCD simulations for the *ab initio* derived quartic force field are performed. The vibrational frequencies obtained from its power spectrum are good agreement with those obtained by accurate methods such as VPT2 and VCI.

The second is the proton transfer reactions in model DNA base pairs. We numerically showed the geometric isotope effects on the stability of the proton-transferred structures of the DNA base pairs as a function of the mass. We performed QCD simulations in order to investigate dynamical stability of the proton-transferred GC pair. The results showed that the proton-transferred structure of the protonated isotopomer is dynamically unstable and that of deuterated isotopomer remains stable. In former case, dynamically induced transi‐ tion from the metastable to global minimum occurs. It is relevant to include dynamical ef‐ fects to treat quantum isotope effects on the proton transfer reactions.

The last application is structural transition of finite quantum Morse clusters. We first com‐ pared the energy of the stable structures of the classical M*n* cluster with those of quantum counterpart and found that the quantum effects due to zero point vibration is remarkable for small system and suppressed for larger. Then we performed the real-time dynamics to evaluate the Lindemann index to characterize the dynamical effects on the melting for M7 cluster. In between solid-like and liquid-like phases (so-called coexistence phase), structural changes of the cluster occur intermittently.

### **Acknowledgements**

**Figure 3.** Static Lindemann indexes evaluated from QCD quantum distance (red), QCD classical distance (blue), and

As an extension to the mechanics concerning about Ehrenfest theorem, we formulated a quantal cumulant mechanics (QCM) and corresponding dynamic method (QCD). The key point is the use of a position shift operator acting on the potential operator and introducing the cumulant variables to evaluate it, so that one need not truncate the potential, and it does not require separating into quantum and classical parts. In particular, we derived the cou‐ pled equation of motion (EOM) for the position, momentum, and second-order cumulants of the product of the momentum and position fluctuation operators. The EOM consists of variables and a quantal potential and its derivatives, where the quantal potential is ex‐ pressed as an exponential function of the differential operator acting on the given potential. We defined density and joint density evaluated from the cumulant expansion scheme. It is clearly found that the present second-order approach gives a Gaussian density distribution spanned both on position and momentum space. Since the density is normalized, the joint density is considered exactly as probability distribution. We also indicated the relation be‐ tween the joint density and cumulant variables as expectation values calculated from the distribution. We extended the QCD for the one-dimensional system in to treat the multi-di‐

mensional systems. We derived the EOMs with the 24*N* dimensional phase space.

CD classical distance (green).

312 Advances in Quantum Mechanics

**4. Summary**

This study is supported by a Grant-in-Aid for Young Scientists (A) (No. 22685003) from Ja‐ pan Society for the Promotion of the Science (JSPS) and also by a CREST program from Ja‐ pan Science and Technology Agency (JST).

### **Author details**

Yasuteru Shigeta1,2

1 Department of Materials Engineering Science, Graduate School of Engineering Sciences, Osaka University, Machikaneyama-cho, Toyonaka, Osaka, Japan

2 Japan Science and Technology Agency, Kawaguchi Center Building, Honcho, Kawaguchishi, Saitama, Japan

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Quantal Cumulant Mechanics as Extended Ehrenfest Theorem

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**Chapter 14**

**Provisional chapter**

**Unruh Radiation via WKB Method**

**Unruh Radiation via WKB Method**

Additional information is available at the end of the chapter

through a one dimensional step potential) and its applications.

probability for the particle to tunnel through the barrier.

Quantum mechanics has many features which are distinct from classical physics. Perhaps none more so than tunneling – the ability of a quantum particle to pass through some potential barrier even when, classically, it would not have enough energy to do so. The examples of tunneling phenomenon range from the nuclear (e.g. alpha decay of nuclei) to the molecular(oscillations of the ammonia molecule). Every text book on quantum mechanics devotes a good fraction of of page space to tunneling (usually introduced via the tunneling

In general, tunneling problems can not be solved, easily or at all, in closed, analytic form and so one must resort to various approximation techniques. One of the first and mostly useful approximations techniques is the WKB method [1] names after its co-discovers Wentzel, Krammers and Brillouin. For a particle with an energy *E* and rest mass *m* moving in a one-dimensional potential *V*(*x*) (where *E* < *V*(*x*) for some range of *x*, say *a* ≤ *x* ≤ *b*, which is the region through which the particle tunnels) the tunneling amplitude is given by

where *p*(*x*) is the canonical momentum of the particle. Taking the square of (1) gives the

In this chapter we show how the essentially quantum field theory phenomenon of Unruh radiation [4] can be seen as a tunneling phenomenon and how one can calculate some details of Unruh radiation using the WKB method. Unruh radiation is the radiation seen by an observer who accelerates through Minkowski space-time. Via the equivalence principle (i.e. the local equivalence between observations in a gravitational field versus in an accelerating

<sup>=</sup> exp

©2012 Singleton, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 A. Singleton; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 A. Singleton; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

−1 *h*¯ *<sup>b</sup> a*

*p*(*x*)*dx*

, (1)

[2*m*(*V*(*x*) <sup>−</sup> *<sup>E</sup>*)]1/2*dx*

Additional information is available at the end of the chapter

Douglas A. Singleton

Douglas A. Singleton

10.5772/53898

**1. Introduction**

http://dx.doi.org/10.5772/53898

exp −1 *h*¯ *<sup>b</sup> a*


### **Chapter 14**

**Provisional chapter**
