1. Introduction
Since Schrödinger proposed wave mechanics for quantum phenomena in 1926 [14], referred as Schrödinger equation named after his name, this equation has been applied to atommolecules, condensed matter, particle, and elementary particle physics and succeeded to reproduce 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 Schrödinger equation depends strongly on the quality of the trial wave function. He also derived the timedependent Schrödinger equation by imposing the timeenergy correspondence. This extension opened to describe timedependent phenomena within quantum mechanics. However there exist a few exactly solvable systems so that the methodology to solve Schrödinger equation approximately is extensively explored, yet.
In contrast to the timedependent wave mechanics, Heisenberg developed the equations of motion (EOM) derived for timedependent operator rather than wave function [5]. This equation is now referred as the Heisenberg’ EOM. This equation is exactly equivalent to the timedependent 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 onedimensional systems and an extension to multidimensional cases is derived. Then, we show three applications in molecular 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 timedependent of an arbitrary operator
The Heisenberg’ equation of motion (EOM) is given as
where
For onedimensional case, the Hamiltonian operator is expressed as a sum of the kinetic and the potential operator as
The Heisenberg’ EOMs for both a coordinate and a momentum are derived as
These equations resemble corresponding Newton’ EOMs as
This relationship is socalled Ehrenfest’s theorem [6]. A definite difference between Heisenberg’ and Nowton’ EOMs is that the expectation value of the potential operator appears in the former. If one approximates the expectation value as
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 referred as the Ehrenfest’s theorem.
In general, Taylor expansion of the potential energy term,
gives a infinite series of higherorder derivatives,
The first term appears in Eq. (27) 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 higherorder moment.
2.2. Quantized Hamilton dynamics and quantal cumulant dynamics
Ehrenfest’ theorem fulfills for the arbitrary wave function. In previous studies, effects of the higherorder moments on dynamics were explored. The most of studies treat secondorder term with the potential being a series of
Recently Shigeta and coworkers derived a general expression for the expectation value of an arbitrary operator by means of cumulants rather than moments [919]. For onedimensional case, the expectation value of a differential arbitrary operator,
where we introduced the general expression for the cumulant
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 secondorder cumulant appears as
and the other higherorder terms do not.
Up to the secondorder, Heisenberg’ EOMs for cumulants are given by
where
It is noteworthy that the first and second terms of above equation corresponds to the first and second terms of Eq. (29), 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. (214). By using the famous formula for the Gaussian integral
the exponential operator appearing in Eq. (214) is rewritten as,
The first derivative operator term in right hand side of the above equation can act to the potential with the relationship of
Therefore it is possible to estimate potential energy term without the truncation of the potential. However the analytic integration is not always has the closed form and the numerical integration does not converge depending on the kind of the potential. For the quantal potential including third and higherorder culumant, it is convenient to use the Fourier integral 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. (213), there exists first integral that always hold for. Now defining a function,
and differentiating it result in
Thus, this function is a timeindependent constant. It is wellknown that the least uncertainty state fulfills
Now by considering the dimension we define new coordinate and momentum as
The secondorder momentum cumulant
Total energy are expressed using the cumulant variables as
Above expression indicates that the energy does not depend on
By means of the new coordinate and momentum, the total energy is rewritten as
This equation tells us that the effective potential derived from the kinetic energy term affect the dynamics of
gives stationary state that fulfills the least uncertainty condition as
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 condition gives a set of solutions as
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 secondorder cumulant variables. Now the density finding a particle at
Thus the secondorder expression for the density is evaluated as
This density shows that the distribution is a Gaussian centered at
This expression indicates that the expectation value of the potential is related to the mean average of the potential with weight
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,
The secondorder expression is given 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
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 joint distribution fulfills the following relations as
Moreover the coordinate, momentum, and cumulants are derived by means of the joint distribution as
2.5. Extension to multidimensional systems
The Hamiltonian of an
where
the total energy is derived as an extension of Eq. (226) by
where
where
From Heisenberg EOM, EOMs up to the secondorder cumulants are given by
where
In contrast to the onedimensional problems, secondorder cumulants are represented as matrices. Thus, the total degrees of freedom are 24
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 molecular 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
where
By taking each normal mode as the degree of freedom in the dynamics simulation, the Hamiltonian for QCD2 with
where
For molecules with more than 1 degree of freedom, we applied 3MRQFF, because it has been shown by various examples that the 3MRQFF is sufficient to describe fundamental modes as well as more complex overtone modes. The QCD2 and classical simulations were performed numerically with a fourthorder RungeKutta integrator. For formaldehyde (CH_{2}O) and formic acid (HCOOH), 3MRQFF PES was generated at the level of MP2/augccpVTZ [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 selfconsistent field method (VSCF) with full secondorder 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 Fourier transform of any dynamical variables obtained from the trajectories of MD simulations is related to spectral densities. In particular, Fourier transform of velocity autocorrelation function gives the density of vibrational states. In addition, the power spectrum of the time series 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 trajectory is obtained, the resolution of the Fourier spectrum becomes fine.
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.







H_{2}CO  ν_{1}  3040  2901  2843  2866  2843 
ν_{2}  2997  2868  2838  2849  2782  
ν_{3}  1766  1764  1723  1734  1746  
ν_{4}  1548  1504  1509  1515  1500  
ν_{5}  1268  1247  1250  1251  1250  
ν_{6}  1202  1166  N/A  1189  1167  
HCOOH  ν_{1}  3739  N/A  3527  3554  3570 
ν_{2}  3126  N/A  2980  2989  2943  
ν_{3}  1794  N/A  1761  1761  1770  
ν_{4}  1409  N/A  1377  1385  1387  
ν_{5}  1302  N/A  1270  1231  1229  
ν_{6}  1130  N/A  1120  1097  1105  
ν_{7}  626  N/A  631  620  625  
ν_{8}
/ν_{9} 
1058 /676 
N/A  N/A  1036 /642 
1033 /638 
3.2. Proton transfer reaction in guaninecytosine base pair
DNA base pairs have two and three interbase hydrogen bonds for AdenineThymine and GuanineCytosine pairs, respectively. Proton transfer reactions among based were theoretically investigated by quantum chemical methods and further quantum mechanical analyses for decases [2833]. In order to investigate dynamical stability of protontransferred structures of the model system consisting DNA bases, we here perform QCD2 simulations of a model GuanineCytosine base pair. The model potential is given by
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/ccpVDZ). The reaction coordinates
with
where Greek characters denote the cumulant variables. In order to avoid the particles escaping from the bottoms, we have added the welllike potential is defined as
where
where
In the actual calculations, the time interval used was 0.1fs, total time is 2ps. The initial conditions of the variables can be determined by the least quantal energy principle. In figures 2 we have depicted phase space (
3.3. Quantal structural transition of finite clusters
Melting behavior of the finite quantum clusters were extensively investigated by many researchers using different kind of methodologies [3439]. We here investigate the melting behavior of
where
which has an analytic form of the quantal potential and coefficients
We here evaluate optimized structures of M_{n} clusters (n=37) for
For the analyses on quantum melting behavior, the parameters of the Morse potential are chosen as
In order to measure the melting behavior of the finite clusters, we here use the Lindemann index defined as
where
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 increases as the increase of the additional kinetic energy. In this region, the structural transition does not actually occur and the cluster remains stiff. This phase is called “solidlike phase”. On the other hand, above a melting point, the structural transition often occurs and the cluster is soft. This phase is called “liquidlike phase”. Between two phases, the Lindemann index rapidly increases. This phase is referred as “coexistence phase”, which is not allowed 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 zeropoint vibrational energy is included 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 socalled quantum softening as indicated by Doll and coworker for the Neon case by means of the pathintegral approach. Our realtime dynamics well reproduce their tendency for this static property. On the other hand, behavior of Lidemann indexes from











2.52135  2.55978  2.56510  3.00000  1.07198  1 

5.20344  5.22683  5.24237  6.0000  1.07811  1 

8.71392  8.74464  8.76049  9.85233  0.99212 1.15085 
0.91725 1.08364 

13.1836  13.2141  13.2141  14.7182  1.00473 1.37518 
0.93581 1.32343 

18.4016  18.4404  18.4415  20.3282  0.88288 1.01263 
0.81221 0.948237 
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 coupled equation of motion (EOM) for the position, momentum, and secondorder 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 expressed 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 secondorder 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 between the joint density and cumulant variables as expectation values calculated from the distribution. We extended the QCD for the onedimensional system in to treat the multidimensional systems. We derived the EOMs with the 24
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
The second is the proton transfer reactions in model DNA base pairs. We numerically showed the geometric isotope effects on the stability of the protontransferred structures of the DNA base pairs as a function of the mass. We performed QCD simulations in order to investigate dynamical stability of the protontransferred GC pair. The results showed that the protontransferred structure of the protonated isotopomer is dynamically unstable and that of deuterated isotopomer remains stable. In former case, dynamically induced transition from the metastable to global minimum occurs. It is relevant to include dynamical effects to treat quantum isotope effects on the proton transfer reactions.
The last application is structural transition of finite quantum Morse clusters. We first compared 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 realtime dynamics to evaluate the Lindemann index to characterize the dynamical effects on the melting for M_{7} cluster. In between solidlike and liquidlike phases (socalled coexistence phase), structural changes of the cluster occur intermittently.
Acknowledgments
This study is supported by a GrantinAid for Young Scientists (A) (No. 22685003) from Japan Society for the Promotion of the Science (JSPS) and also by a CREST program from Japan Science and Technology Agency (JST).References
 1.
Schrödinger, E. (1926). Quantisierung als Eigenwertproblem (Erste Mitteilung). Annalen der Physik, Vol. 79, No. 4, (April, 1926), pp. 361376. ISSN: 00033804  2.
Schrödinger, E. (1926). Quantisierung als Eigenwertproblem (Zweite Mitteilung). Annalen der Physik, Vol. 79, No. 6, (May, 1926), pp. 489527. ISSN: 00033804  3.
Schrödinger, E. (1926). Quantisierung als Eigenwertproblem (Dritte Mitteilung: Storungstheorie, mit Anwendung auf den Starkeffekt der Balmerlinien). Annalen der Physik, Vol. 80, No. 13, (September, 1926), pp. 437490. ISSN: 00033804.  4.
Schrödinger, E. (1927). Quantisierung als Eigenwertproblem (Vierte Mitteilung). Annalen der Physik, Vol 81, No. 18, (September, 1927), pp. 109139. 00033804.  5.
Heisenberg, W. (1943). The observable quantities in the theory of elementary particles. III. Zeitschrift für physik, Vol 123, No. 12, (March, 1943) pp. 93112. ISSN: 00443328.  6.
Ehrenfest, P. (1927). Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik. Zeitschrift für physik, Vol 45, No. 78, (July, 1927), pp. 455472. ISSN: 00443328.  7.
Prezhdo, O.V. & Pereverzev, Y.V. (2000). Quantized Hamilton dynamics. Journal of Chemical Physics, Vol. 113, No. 16, (October 22, 2000), pp. 65576565. ISSN: 00219606.  8.
Prezhdo, O.V. (2006). Quantized Hamilton Dynamics. Theoretical Chemistry Accounts, Vol. 116, No. 13, (August 2006), pp. 206218. ISSN: 1432881X and references cited therein.  9.
Miyachi, H.; Shigeta, Y.; Hirao K. (2006). Real time mixed quantumclassical dynamics with ab initio quartic force field: Application to molecular vibrational frequency analysis. Chemical Physics Letters, Vol. 432, No. 46, (December 11, 2006) 585590. ISSN: 00092614  10.
Shigeta, Y.; Miyachi, H.; Hirao, K. (2006). Quantal cumulant dynamics: General theory. Journal of Chemical Physics, Vol. 125: 244102. ISSN: 00219606.  11.
Shigeta, Y.; Miyachi, H.; Hirao, K. (2007). Quantal cumulant dynamics II: An efficient timereversible integrator. Chemical Physics Letters, Vol. 443, No. (AUG 6 2007), 414419. ISSN: 00092614  12.
Shigeta Y. (2007). Quantal Cumulant Dynamics for Dissipative Systems. AIP proceedings Vol. 963, (2007), 1317.  13.
Shigeta Y. (2008). Quantal cumulant dynamics III: A quantum confinement under a magnetic field. Chemical Physics Letters, Vol. 461, No. 46, (August 20, 2008), 310315. ISSN: 00092614  14.
Shigeta Y. (2008). Distribution function in quantal cumulant dynamics. Journal of Chemical Physics, Vol. 128, No. 16, (April 28, 2008) 161103. ISSN: 00219606.  15.
Shigeta, Y.; Miyachi, H.; Matsui, T.; Hirao, K. (2008) Dynamical quantum isotope effects on multiple proton transfer reactions. Bulletin of the Chemical Society Japan. Vo. 81, No. 10, (October 15, 2008), 1230 1240. ISSN: 00092673.  16.
Pereverzev, Y.V.; Pereverzev, A.; Shigeta, Y.; Prezhdo, O.V. (2008) Correlation functions in quantized Hamilton dynamics and quantal cumulant dynamics. Journal of Chemical Physics, Vol. 129, No. 14, (October 14, 2008), 144104. ISSN: 00219606.  17.
Shigeta, Y. Molecular Theory Including Quantum Effects and Thermal Fluctuations, the Bulletin of Chemical Society Japan, Vo. 82, No. 11, (November 15, 2009), 13231340. ISSN: 00092673.  18.
Shigeta, Y.; Miyachi, H.; Matsui, T.; Yokoyama, N.; Hirao, K. "Quantum Theory in Terms of Cumulant Variables", Progress in Theoretical Chemistry and Physics, Vol. 20, "Advances in the Theory of Atomic and Molecular Systems  Dynamics, Spectroscopy, Clusters, and Nanostructures", edited by Piecuch, P.; Maruani, J.; DelgadoBarrio, G.; Wilson, S., pp. 334, Springer, 2009, 334.  19.
Shigeta, Y.; Inui, T.; Baba, T.; Okuno, K.; Kuwabara, H.; Kishi, R.; Nakano, M. (2012) International Journal of Quantum Chemistry. in press, (2012). ISSN: 00207608.  20.
Mayer, J. E. (1937). The Statistical Mechanics of Condensing Systems I. Journal of Chemical Physics, Vol. 5, (January, 1937), 6773. ISSN: 00219606.  21.
Kubo, R. (1962). Generalized Cumulant Expansion Method. Journal of Physical Society of Japan, Vol. 17, No. 7, (July, 1962) 11001120. ISSN: 00319015.  22.
Mandal. S.H.; Sanyal, G.; Mukherjee, D. (1998). A Thermal ClusterCumulant Theory. Lecture Notes in Physics, Vol. 510, (1998), 93117. ISSN: 00758450.  23.
Yagi, K.; Hirao, K.; Taketsugu, T.; Schmidt, M.W.; Gordon, M.S. (2004). Ab initio vibrational state calculations with a quartic force field: Applications to H_{2}CO, C_{2}H_{4}, CH_{3}OH, CH_{3}CCH, and C_{6}H_{6}.Journal of Chemical Physics , Vol. 121, No. 3 (July 15, 2004) 13831389. ISSN: 00219606.  24.
Möller, C; Plesset, M.S. (1934). Note on an approximation treatment for manyelectron systems. Physical Review, Vol. 46, No. 7, (October, 1934), 618622. ISSN: 0031899X.  25.
Lendall, R.A.; Dunning Jr, T.H.; Harrison, R.J. (1992). Electron affinities of the firstrow atoms revisited. Systematic basis sets and wave functions. Journal of Chemical Physics , Vol. 96, No. 9, (May 1, 1992), 67966806. ISSN: 00219606.  26.
M.W. Schmidt et al (1993) General Atomic and Molecular Electronicstructure System. Journal Computational Chemistry , Vol. 14, No. 11, (November, 1993), 13471363. ISSN: 01928651.  27.
M.J. Frisch et al (2004) Gaussian 03 (Revision C.02). Gaussian Inc Wallingford CT  28.
Florián, J.; Hrouda, V.; Hobza, P. (1994). Proton Transfer in the AdenineThymine Base Pair. Journal of the American Chemical Society, Vol. 116, No. 4, (February 23, 1994), 14571460. ISSN: 00027863.  29.
Floriaán J, Leszczyn’sky J (1996) Spontaneous DNA Mutation Induced by Proton Transfer in the GuanineCytosine Base Pair: An Energetic Perspective. Journal of the American Chemical Society, Vol. 118, No. 12, (March 27, 1996), 30103017. ISSN: 00027863.  30.
Villani G (2005) Theoretical investigation of hydrogen transfer mechanism in adeninethymine base pair. Chemical Physics , Vol. 316 No. 13, (September 19, 2005), 18. ISSN: 03010104.  31.
Villani G (2006) Theoretical investigation of hydrogen transfer mechanism in the guaninecytosine base pair Chemical Physics, Vol. 324, No. 23, (MAY 31 2006), 438 446. ISSN: 03010104.  32.
Matsui T, Shigeta Y, Hirao K (2006) Influence of Pt complex binding on the guanine cytosine pair: A theoretical study. Chemical Physics Letters , Vol. 423, No 46, (June 1, 2006), 331334. ISSN: 00092614.  33.
Matsui, T.; Shigeta, Y.; Hirao, K. (2007). Multiple protontransfer reactions in DNA base pairs by coordination of Pt complex. Journal of Chemical Physics B , Vol. 111, No. 5, (February 8, 2007), 11761181. ISSN: 15206106.  34.
Ceperley, D.M. (1995). Pathintegrals in the theory of condensed Helium. Review Modern Physics , Vol. 67, No. 2, (April 1995), 279355. ISSN: 00346861.  35.
Chakravarty, C. (1995). Structure of quantum binary clusters. Physical Review Letters , Vol. 75, No. 9, (August 28, 1995), 17271730. ISSN: 00319007.  36.
Chakravarty, C. (1996). Cluster analogs of binary isotopic mixtures: Path integral Monte Carlo simulations. Journal of Chemical Physics, Vol. 104, No. 18, (May 8, 1996), 72237232. ISSN: 00219606.  37.
Predescu, C.; Frantsuzov, P.A.; Mandelshtam, V.A. (2005). Journal of Chemical Physics , Vol. 122, No. 15, (April 15, 2005), 154305. ISSN: 00219606.  38.
Frantsuzov, P.A.; Meluzzi, D.; Mandelshtam, V.A. (2006). Structural transformations and melting in neon clusters: Quantum versus classical mechanics. Physical Review Letters , Vol. 96, No. 11, (March 24, 2006), 113401. ISSN: 00319007  39.
Heller, E. J. (1975). Timedependent approach to semiclassical dynamics. Journal of Chemical Physics , Vol. 62, No. 4, (1975), 15441555. ISSN: 00219606.