- Email: [email protected]
Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach
Accepted Manuscript Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach Mohamad Toutounji PII: DOI: Reference:
S0003-4916(14)00304-2 http://dx.doi.org/10.1016/j.aop.2014.10.010 YAPHY 66642
To appear in:
Annals of Physics
Received date: 10 September 2014 Accepted date: 15 October 2014 Please cite this article as: M. Toutounji, Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach, Annals of Physics (2014), http://dx.doi.org/10.1016/j.aop.2014.10.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Reconsidering Harmonic and Anharmonic Coherent States: Partial Differential Equations Approach
Mohamad Toutounji College of Science, Department of Chemistry, P. O. Box 15551, UAE University, Al-Ain, UAE [email protected] ABSTRACT
This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron-phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.
1
I.
Introduction
Probing time development of anharmonic systems for acquiring dynamical and structural information can be challenging, especially when Morse potential is employed for accounting for the anharmonic character of the system of interest. Besides the difficult algebraic nature of Morse potential, Morse oscillator tends to display divergent dynamics.1-5Additionally, dynamical calculations that involve utilizing Morse oscillator eigenfunctions suffer from numerical instabilities and convergence issues. Furthermore, essential singularity issues crop up in the classical limit upon using Morse Hamiltonian. A long standing problem in Morse oscillator dynamics or statistical mechanics has been the divergence that arise upon evaluating average values and time correlation functions classically4 or quantum mechanically.1,5 This particular problem has been elucidated by Toutounji in detail.)2-4 The above difficulties associated with using Morse oscillator have always motivated research groups to come up with techniques to surmount those difficulties. Very recently, Toutounji6 has used Morse coherent states as an alternative to using Morse oscillator eigenstate representation or its corresponding propagator. Coherent states (harmonic or anharmonic) can be very demanding algebraically. The difficulty traces back to the noncommuting operators involved in coherent states manipulation. Besides, unfamiliar expressions involving operator algebra which often lead to infinite series might crop up,while using coherent states, that is not easily summable to a finite value.7,8 The main focus of this article is to present a new approach for evaluating time-dependent variables that are essential components to spectroscopy and quantum dynamics, e.g. linear/nonlinear dipole moment time/frequency correlation function, position correlation function, quantum solvation, wavepacket dynamics, scattering, etc. Theses quantities require time evolution operator acting on the state function of the system of interest. Morse oscillator has been well utilized in modeling anharmonic molecular vibrations for which the reason it will be used herein as the system of interest for probing anharmonic nuclear dynamics. Different groups have expended efforts on developing anharmonic ladder operators for the purpose of constructing of Morse oscillator coherent states,9-15 but not much on their utility and the corresponding operations of its coherent states in dynamics. For example, Popov and coworkers, and others, have reported important work on Morse oscillator coherent states, however their work was missing utility and operational properties of Morse coherent states.9-11 The motivation for our work is the recent article by Popov et al15 where their results were reported in terms of sums and exponential derivatives. It turns out that employing partial differential equations approach can help eliminate the sums and derivatives and recast the results in a closed form. This approach has made us realize that manipulating coherent states may be reconsidered using the aforementioned approach, leading to more manageable results, including harmonic coherent states which will be used here for ratifying and illustrative purposes. However, our main focus here will be solving problems related to Morse coherent states and the 2
associated spectroscopy and dynamics such as evaluating time-correlation functions. This method has come to our attention during the course of the work that had led to the work in Ref. 6. Toutounji8 had used a system of first-order linear ordinary differential equations to solve noncommutative algebra related problems, whereby linear and quadratic electron-phonon coupling in condensed phase systems was treated. Recently, Popov et al10,11,15 and Lemus16 have done remarkable work on the algebraic structure of Morse coherent states and dynamical related issues. Most notable work in15 is Morse thermal states which we intend to explore using the herein approach. In this article, the equilibrium canonical partition function and non-equilibrium autocorrelation function of harmonic oscillator are reproduced analytically using partial differential equations approach instead of manipulating harmonic coherent states algebraically to ratify the correctness and applicability of the presented approach. Similarly, the corresponding functions of Morse oscillator are derived using the same differential approach, rather than the algebraic one. II.
Harmonic Oscillator Thermal States and Quantum Dynamics
Although the results provided in this section are well known in the literature, they are reproduced for illustrative and ratifying purposes of the approach presented herein. While the first part of this section reproduces the equilibrium canonical partition function of a quantum harmonic oscillator, the second part illustrates the case for autocorrelation function (used in scattering) employing differential equations. A. The equilibrium canonical partition function The equilibrium canonical partition function may be defined as Tre ,
(1)
e / | .
(2)
where is the harmonic oscillator Hamiltonian operator and β is the inverse Boltzmann constant. Upon expanding the quantum trace in a complete basis set such as the harmonic coherent states |〉 in Eq. 1, we obtain
We shall assume that the above coherent states are un-normalized for convenience. integrand in Eq. 2 may be written as
!"#/$ %
e "& ' (/(' / | . 3
The
(3)
Before evaluating the integral, one needs to re-express the integrand in Eq. 3 in a more manageable form. We are interested in how the exponential operator e "&' (/(' acts on some arbitrary function )*+, assuming it is a physically well-behaved function. If we let
e, '-. )*+ /*0, +, -
(4)
consider the following partial differential equation (PDE) (
(,
/ *0, +
(
(2
/ *0, +
(5)
with the initial condition at 0 0, /*0, + )*+. The differential equation in Eq. 5 is firstorder linear homogeneous PDE and can be solved using the method of characteristics,17 yielding /*0, + /*4 5 +,
(6)
49:;|| e "& 6 || <.
(7)
csc *β"ω/2+.
(8)
where 0 6β"ω. Using the solution in Eq. 6 in Eq. 3, we get
!"#/$ %
Carrying out the integral in Eq. 7 yields the harmonic oscillator partition function
B. Dynamical Autocorrelation function The general expression for autocorrelation function of any system is defined as inner product of initial state of wavepacket Ψ(0) and its final state evolution Ψ(t)18,19 Ψ*A+|Ψ*0+ ∑D|CD | exp *6GHD A+
*10+
These probabilities |CD | may be viewed as Franck-Condon factors for the sake of the forthcoming calculations that will be carried out herein, HD *J K 1/2+"ω. In this section an exact expression for the autocorrelation function of harmonic oscillator using both coherent states and the above PDE approach will be derived. Starting with the vacuum state |0〉, our initial wavepacket is excited to the electronic excited state where it evolves under the nuclear 4
force of that state. Starting with the initial wave packet in the ground state and ending with the final state eLMNO/" |0 , the dynamics may be represented by the autocorrelation function PQ *A+ PQ *A+ 0eLMNO/" 0.
(11)
The final state is represented by a linearly displaced harmonic oscillator, keeping the same harmonic frequency as in the ground state. The ground and excited nuclear Hamiltonians (expressed in dimensionless quantities) are ℏω 2 P + q2 ) ( 2
(12)
ℏω 2 P + (q + ∆ )2 ) ( 2
(13)
Hg =
He =
where ∆ is the upper linear dimensionless displacement with respect to the initial state. Introducing the following unitary transformation, eLMNO/" exp R6G*ST U S+A/"V,
(14)
which allows avoiding dealing directly with the excited nuclear Hamiltonian. S in Eq. 14 is the unitary translational operator. Inserting Eq. 14 in Eq. 11 leads to PQ *A+ W
∆
√
ZeLM[O/" Z
∆
\.
√
(15)
Constructing similar PDE to that in Eq. 5 and solving it using the method of characteristics17 leads to ∆2 iω t Fh (t ) = exp − (1 − e − iωt ) − . 2 2
(16)
The result in Eq. 16 is in exact agreement with the autocorrelation function of displaced harmonic oscillator. III.
Morse Oscillator Thermal States and Quantum Dynamics
In this section we will apply the approach of PDE to reproduce the equilibrium canonical partition function of Morse oscillator. We will also apply the same approach to evaluate the 5
optical time autocorrelation function of the same anharmonic oscillator. The Morse oscillator Hamiltonian assumes this form (expressed in dimensionless quantities)
H=
(
ℏω 2 1 P + 1 − e− 2 2χ
2χq
) , 2
*17+
where χ is the anharmonicity constant. A. The equilibrium canonical partition function
While |〉 was used in the previous section to denote harmonic coherent states, |^〉 will be used here to signify Morse coherent states. It is therefore useful to provide a brief account of Morse oscillator coherent states. Morse coherent states may be defined as eigenstates of the lowering operator _ (sometimes referred to as annihilation operator) as _|^〉 z|^〉
*18+
Un-normalized Morse coherent states written as a linear combination of the Morse oscillator eigenstates |J〉as |^〉 ∑f/
bc Dgh de*D+ |J〉
*19+
where ρ(m) will be obtained by using the definition of Klauder-Perelomov coherent states, namely |^〉 49:*^_T +|0〉,
(20)
where _T is the anharmonic raising operator and |0〉 is the vacuum state of Morse oscillator. The quantum number m in Eq. 19 signifies the number of anharmonic phonons with m = 0, 1, …,
[N/2], where j k"$ 2$N 6 1, with o and being the reduced mass and the Morse parameter lmn
which governs the breadth of Morse potential, respectively. Expanding the above exponential in Taylor series and carrying out the _T operation yields11 6
p*J+
qr s*Dt+s*fDt+ s*ft+
(21)
Note that coherent states in Eq. 19 were left un-normalized for convenience and better book keeping. It is also helpful at this point to define the ladder operators for Morse oscillator, _k
"u nN
_T k
v
"u nN
nN
"u
v
nN
"u
6
K
w x
xw
w x
6 z,
xw
w y
6 z, w y
(22)
(23)
where the Morse coordinate { *4S} /"~+4 2 with S} , ~, and being the dissociation energy, harmonic frequency, and the Morse parameter, respectively. Assuming that Morse potential will have [N/2] finite bound states, the inner product of two coherent states |^ 〉 and |^ 〉 is ^ |^ ∑Rf/V Dgh
*b ∗ b$ +c e*D+
*24+
One would need the closure relationship in order to evaluate time correlation functions. One may start with the projection operator (assuming the system of interest escapes dissociation state so as to warrant completeness of the vibrational bound states) ∑Rf/V Dgh |J J |
(25)
which will lead to the following closure relation in terms of Morse coherent states (unnormalized) |^ ^| ,
(26)
where the form of the integration measure dσ that I have used in Eq. 26 is missing the normalization factor since I have chosen my coherent states to be un-normalized,
7
*ft+
$ ||$ f t
^.
(27)
The equilibrium canonical partition function of Morse oscillator may be expanded in terms of its coherent states as6
*ft+ f
b} b
$ ||$ t
^
(28)
where d2z = dxdy, with ^ 9 K G{. One may also express z in polar coordinates as ^ 04 . For the Morse oscillator the exponential Boltzmann operator may be recast as e |^〉 49: 6 vHh K ^
b
6 ^
$
b $
z |^〉,
(29)
where *1 6 +"~ and "~ . Hh is the zero-point energy of the Morse oscillator. The
differential operators ^ b and ^ b $ commute for which the reason they may be utilized interchangeably. Acting successively with these two operators Eq. 29 becomes
$
e |^〉 49:R6Hh V 49: 6^
b
49: ^
$
b $
|^〉.
(30)
Proceeding along the same lines as in Sec. II, two PDEs along with their initial conditions will be obtained. Assuming that our physical function that needs to be operated on with exp *^ b+ is
Ψ(η, z) while for the second order exponential operator exp *^ b $ + it is G*θ, z+, then the firstand second order (parabolic) partial differential equations (PDE) need to be solved are $
and
Ψ*, ^+ ^
b
Ψ*, ^+
(31)
G*, ^+
(32)
Ψ*0, ^+ Φ*^+
G*, ^+ ^
$
b $
G*0, ^+ *^+. 8
Utilizing the method of characteristics, the PDE in Eq. 31 is readily solved to yield Ψ*, ^+ Φ*4 ^+.
(33)
Ψ*, ^+ ∑∞ gh *+^ .
(34)
f/ Ψ*, ^+ h K z K ∑g exp * * 6 1++^ ,
(35)
While the parabolic PDE in Eq. 32 may conventionally be transformed into the heat/diffusion equation using Laplace transform, it was found easier to use the power series method directly
to find
where and are expansion coefficients. Now equating G*0, ^+ to the inner product of Morse coherent states, by taking *^+ ^|^ , will immediately render the final solution, f/ Γ*ft+ b ∗ 49:* !*f+! f
Ψ*, ^+ ∑gh
* 6 1++^ .
Using Eqs. 33−36 leads to
49: 6 vH0 K ^
4 ∑gh
^
6 ^2
f/ Γ*ft+ b ∗ f 49:* !*f+!
2
^2
z *^+
* 6 1++^4
(36)
(37)
Alternatively, one can get a closed-form expression using series expansion, leading to 49: ¡6 ¢Hh K ^ |^| £ ¢1 K j
f
6 ^ £¤ *^+ ^ ^
f⁄
j K |^| 4 2|^| 4 ¢2 K £ ¥ ∗ ¦ j ^ *^4 + 9
f
× U v6 , , yb ∗ $ f
z *b} © +$
*ft|b|$ } © +$
where *^+ ^|^ and °*, ±, ^+ is the confluent hypergeometric function of the second kind.20
*38+
Using the above approach, one can take a closer look at the thermal states of Morse oscillator derived recently by Popov et al15 and obtain a more direct closed-form rather than leaving it in a differential form. The reported form in Ref. 15 reads as 4 c 49:R6Hh V 49: $ 49:R6JV.
(39)
²*C, + 49: $ 49:R6JV.
(40)
$
In this case we have $
Proceeding along the same approach established above, one can construct this PDE and its initial value at c = 0
³
² *C, +
$
$
² *C, +
²*0, + ´*+ 49:R6JV,
(41)
where C . The PDE in Eq. 41 is the heat/diffusion equation, on an infinite line in space,
of which solution is
K*C, +
t¶ { √y³ ¶
4 *w+
$ /y³
49:RJ*C J 6 +V.
49:R6J{V
(42)
The result in Eq. 32 offers numerical efficiency that falls out of this sort of stable, smooth Gaussian integrals, which really applies to thermal states of many important systems. It is noteworthy that the treatment of Popov that led to Eq. 39 had imposed an approximation with respect to the number of bound vibrational states, whereas the results reported in Eqs. 37 and 38 bear no approximations and are exact. B. Dynamics and autocorrelation function for weak electron-phonon coupled systems 10
In this section an expression for the autocorrelation function of Morse oscillator using Morse coherent states will be derived. The general formula of the optical autocorrelation function is given in Eq. 10. Starting with the Morse oscillator vacuum state|0〉, the molecular wavepacket makes a transition to the electronic excited state where it evolves under the nuclear force of excited displaced Morse potential. Starting with the initial wave packet in the ground state and ending with the final state eLMNO/" |0 , looking at the Morse oscillator autocorrelation function P*A+ P*A+ 0eLMNO/" 0
(43)
will impart valuable information on the system dynamics. The final state is represented by a linearly displaced Morse oscillator, keeping the same harmonic frequency as in the ground state. The ground and excited nuclear Hamiltonians are
(
Hg =
ℏω 2 1 P + 1 − e− 2 2χ
He =
ℏω 2 1 P + 1 − e− 2 2χ
2χQ
) 2
(44)
(
2 χ (Q −∆ )
) , 2
where ∆ is the upper linear dimensionless displacement with respect to the initial state. eLMNO/" |0 is very hard to carry out algebraically, therefore one may get around that by introducing the following unitary transformation, eLMNO/" exp R6G*ST U S+A/"V,
(45)
S 49:;6∆*_ 6 _T +/√2<,
(46)
∆/√2eLM[O/" ∆/√2.
(47)
which allows avoiding dealing directly with the excited nuclear Hamiltonian. D is a unitary displacement operator and takes on this form
where _ is Morse annihilation operator and _T isthe corresponding creation operator. Inserting Eq. 46 in Eq. 45 leads to
11
∆/√2 is a Morse coherent state and therefore the above relations used to find the partition function will apply. Equation 48 will be evaluated assuming anharmonic weak electron-phonon coupling (conventionally when Huang-Rhys-factor S ≤ 1 where ¸ ¹ /2), leading to compact expressions.
Using the definition of Morse coherent states in Eq. 19 one can show that eLM[O/" |^〉 49: 6G vHh K ^
where *1 6 +"~ and ". autocorrelation function would read P*A+ W
∆
√
b
6 ^
$
b $
z A/" |^〉,
(48)
After utilizing operator algebra techniques the
Z49: 6G vHh K ^
b
6 ^
$
b $
z A/" Z
∆
\.
√
(49)
Developing analogous PDEs as those in Eqs. 31-33 and 38 yields the autocorrelation function for Morse oscillator in the weak electron-phonon coupling limit
f
P*A+ v1 K fz º$
¢2 K
»¼© " º$
f
f
£ ½
$ »¼© L¢ " ftº$ £ "
¾º¿
f⁄
À
° ¢6 , ,
»¼©
f L* " ftº$ +$ "
y¾º¿
£
*50+
where °*, ±, . + is the confluent hypergeometric function of the second kind.20 Careful examination of Eq.50 reveals that the Franck-Condon factor (FCF) for the 0-0 transition is | 0|0 | v1 K z f º$
f
.
(51)
One may recover the FCF for the 0-0 transition of a displaced harmonic oscillator by taking j → ∞, yielding Ä$
| 0|0 | 4 $ .
12
(51)
Equation 51 is the exact FCF documented in the literature. This end, it would be helpful to compare the time correlation function in Eq. 50 to the exact dynamics6 of a system with ~ 50 cm-1, ∆1, and χ 0.0025. Figure 1 shows this calculation where the two curves of Eq. 50 and that of Ref. 6 are identical. IV. Concluding Remarks
The present approach tends to avoid dealing directly with operator algebra and the formidable algebraic techniques needed to manipulate coherent states. The involved algebra gets even more complicated when Morse oscillator coherent states are utilized.12-14 Because the herein approach relies on partial differential equations (most often parabolic if second order), not only does it encompass a broader range of anharmonic potentials, but also offers high numerical efficiency. One can appreciate this by taking a closer look at Morse oscillator eigenfunctions,21 for example. They seem to have numerical instability and convergence issues, for which the reason utilization of Kratzer oscillator was suggested, and discussed in detail, in case of anharmonicity.22 Although using coherent states solves the problem of dealing directly with eigenstate representation, thereby eliminating numerical instability, it introduces another problem−operator algebra and other insurmountable infinite series issues. This paper suggests a PDE approach for dealing with different oscillator coherent states instead of operator algebra manipulation. Harmonic oscillator partition function and autocorrelation function have been derived to ratify the correctness and applicability of the method. Exact agreement of the results shows the success and utility of the method. The utility of this method, though only applied to harmonic and Morse oscillators here, lies in its Gaussian integrals that result upon solving the respective PDEs, especially when modeling requires more complicated potentials. In this case, those integrals feature smoothness and stability. Thus, our approach of utilizing PDEs in solving for quantum dynamics related problems should set the stage for wider applicability in case of more complicated anharmonic systems, as numerical routines are well established for evaluating integrals or solving PDEs. Closed-form expressions for both the partition function and autocorrelation function of Morse oscillator have been derived. In fact, our PDE approach has enabled us to report two exact analytical expressions bearing no restrictions or approximations, e.g. Eqs. 37 and 38.
References
13
1. Wu, J.; Cao, J., J. Chem. Phys. 2001, 115, 5381. 2. Toutounji, M., J. Phys. Chem. C 2010, 114, 20764. 3. Toutounji, M., J. Phys. Chem. B 2011, 115, 5121. 4. Toutounji, M., Phys. Chem. Chem. Phys. 2012, 14, 626. 5. Strekalov, M. L., Chem. Phys. Lett. 2007, 439, 209. 6. Toutounji, M., Theor Chem. Acc. 2014, 133, 1461. 7. Toutounji, M., Chem. Phys. Lett. 2013, 555, 92. 8. Toutounji, M., J. Chem. Phys., 2008, 128, 164103. 9. Popov, D., Phys. Lett. A 2003, 316, 369; EJTP 2006, 11, 123. 10. Daoud, M.; Popov, D. Int. J. Mod. Phys. B 2004, 18, 325. 11. Popov, D.; Zaharie, I.; Dong, S., Czech. J. Phys. 2006, 56, 157. 12. Nieto, M., M.; Simmons, L., M., Phys. Rev. A1979, 19, 438; ibid, Phys. Rev. D 1979, 20,1321; Phys. Rev. D 1979, 20,1342; ibid, Phys. Rev. D 1980, 22, 403. 13. Jellal, A., Mod. Phys. Lett. A 2002, 17, 671. 14. M. Angelova, A. Hertz, V. Hussin, Journal of Physics A: Math. Theor. 2012, 45, 1751. 15. Popov, D.; Dong, S.; Pop, N.; Sajfert, V.; Simon, S., Annals of Physics, 2013, 339, 122. 16. Castanos, O.; Lemus, R., Mol. Phys. 2010, 108, 597. 17. Arfken, G., Mathematical Methods for Physicists; Academic Press, California, 1985. 18. Castlillo, J., F.;Reid, K., L., Laser Chemistry, 1999, 19, 57. 19. Robinett, R. W., Physics Reports, 2004, 392, 1. 20. Gradsteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series and Products; Academic Press: New York, 2007. 21. Toutounji, M., J. Phys. Chem. B 2011, 115, 5121. 22. Toutounji, M., J. Chem. Theory and Compt. 2011, 7, 1804.
14
Figure Caption
Figure 1: Two dipole moment autocorrelation functions of an anharmonic system with ~ 50 cm-1, ∆1, and χ 0.00325 , both of which coincide and mach exactly. While the ordinate takes arbitrary unites, the abscissa is labeled with femtoseconds (fs).
0
1000
2000 Tim e fs
15
3000
S0003-4916(14)00304-2 http://dx.doi.org/10.1016/j.aop.2014.10.010 YAPHY 66642
To appear in:
Annals of Physics
Received date: 10 September 2014 Accepted date: 15 October 2014 Please cite this article as: M. Toutounji, Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach, Annals of Physics (2014), http://dx.doi.org/10.1016/j.aop.2014.10.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Reconsidering Harmonic and Anharmonic Coherent States: Partial Differential Equations Approach
Mohamad Toutounji College of Science, Department of Chemistry, P. O. Box 15551, UAE University, Al-Ain, UAE [email protected] ABSTRACT
This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron-phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.
1
I.
Introduction
Probing time development of anharmonic systems for acquiring dynamical and structural information can be challenging, especially when Morse potential is employed for accounting for the anharmonic character of the system of interest. Besides the difficult algebraic nature of Morse potential, Morse oscillator tends to display divergent dynamics.1-5Additionally, dynamical calculations that involve utilizing Morse oscillator eigenfunctions suffer from numerical instabilities and convergence issues. Furthermore, essential singularity issues crop up in the classical limit upon using Morse Hamiltonian. A long standing problem in Morse oscillator dynamics or statistical mechanics has been the divergence that arise upon evaluating average values and time correlation functions classically4 or quantum mechanically.1,5 This particular problem has been elucidated by Toutounji in detail.)2-4 The above difficulties associated with using Morse oscillator have always motivated research groups to come up with techniques to surmount those difficulties. Very recently, Toutounji6 has used Morse coherent states as an alternative to using Morse oscillator eigenstate representation or its corresponding propagator. Coherent states (harmonic or anharmonic) can be very demanding algebraically. The difficulty traces back to the noncommuting operators involved in coherent states manipulation. Besides, unfamiliar expressions involving operator algebra which often lead to infinite series might crop up,while using coherent states, that is not easily summable to a finite value.7,8 The main focus of this article is to present a new approach for evaluating time-dependent variables that are essential components to spectroscopy and quantum dynamics, e.g. linear/nonlinear dipole moment time/frequency correlation function, position correlation function, quantum solvation, wavepacket dynamics, scattering, etc. Theses quantities require time evolution operator acting on the state function of the system of interest. Morse oscillator has been well utilized in modeling anharmonic molecular vibrations for which the reason it will be used herein as the system of interest for probing anharmonic nuclear dynamics. Different groups have expended efforts on developing anharmonic ladder operators for the purpose of constructing of Morse oscillator coherent states,9-15 but not much on their utility and the corresponding operations of its coherent states in dynamics. For example, Popov and coworkers, and others, have reported important work on Morse oscillator coherent states, however their work was missing utility and operational properties of Morse coherent states.9-11 The motivation for our work is the recent article by Popov et al15 where their results were reported in terms of sums and exponential derivatives. It turns out that employing partial differential equations approach can help eliminate the sums and derivatives and recast the results in a closed form. This approach has made us realize that manipulating coherent states may be reconsidered using the aforementioned approach, leading to more manageable results, including harmonic coherent states which will be used here for ratifying and illustrative purposes. However, our main focus here will be solving problems related to Morse coherent states and the 2
associated spectroscopy and dynamics such as evaluating time-correlation functions. This method has come to our attention during the course of the work that had led to the work in Ref. 6. Toutounji8 had used a system of first-order linear ordinary differential equations to solve noncommutative algebra related problems, whereby linear and quadratic electron-phonon coupling in condensed phase systems was treated. Recently, Popov et al10,11,15 and Lemus16 have done remarkable work on the algebraic structure of Morse coherent states and dynamical related issues. Most notable work in15 is Morse thermal states which we intend to explore using the herein approach. In this article, the equilibrium canonical partition function and non-equilibrium autocorrelation function of harmonic oscillator are reproduced analytically using partial differential equations approach instead of manipulating harmonic coherent states algebraically to ratify the correctness and applicability of the presented approach. Similarly, the corresponding functions of Morse oscillator are derived using the same differential approach, rather than the algebraic one. II.
Harmonic Oscillator Thermal States and Quantum Dynamics
Although the results provided in this section are well known in the literature, they are reproduced for illustrative and ratifying purposes of the approach presented herein. While the first part of this section reproduces the equilibrium canonical partition function of a quantum harmonic oscillator, the second part illustrates the case for autocorrelation function (used in scattering) employing differential equations. A. The equilibrium canonical partition function The equilibrium canonical partition function may be defined as Tre ,
(1)
e / | .
(2)
where is the harmonic oscillator Hamiltonian operator and β is the inverse Boltzmann constant. Upon expanding the quantum trace in a complete basis set such as the harmonic coherent states |〉 in Eq. 1, we obtain
We shall assume that the above coherent states are un-normalized for convenience. integrand in Eq. 2 may be written as
!"#/$ %
e "& ' (/(' / | . 3
The
(3)
Before evaluating the integral, one needs to re-express the integrand in Eq. 3 in a more manageable form. We are interested in how the exponential operator e "&' (/(' acts on some arbitrary function )*+, assuming it is a physically well-behaved function. If we let
e, '-. )*+ /*0, +, -
(4)
consider the following partial differential equation (PDE) (
(,
/ *0, +
(
(2
/ *0, +
(5)
with the initial condition at 0 0, /*0, + )*+. The differential equation in Eq. 5 is firstorder linear homogeneous PDE and can be solved using the method of characteristics,17 yielding /*0, + /*4 5 +,
(6)
49:;|| e "& 6 || <.
(7)
csc *β"ω/2+.
(8)
where 0 6β"ω. Using the solution in Eq. 6 in Eq. 3, we get
!"#/$ %
Carrying out the integral in Eq. 7 yields the harmonic oscillator partition function
B. Dynamical Autocorrelation function The general expression for autocorrelation function of any system is defined as inner product of initial state of wavepacket Ψ(0) and its final state evolution Ψ(t)18,19 Ψ*A+|Ψ*0+ ∑D|CD | exp *6GHD A+
*10+
These probabilities |CD | may be viewed as Franck-Condon factors for the sake of the forthcoming calculations that will be carried out herein, HD *J K 1/2+"ω. In this section an exact expression for the autocorrelation function of harmonic oscillator using both coherent states and the above PDE approach will be derived. Starting with the vacuum state |0〉, our initial wavepacket is excited to the electronic excited state where it evolves under the nuclear 4
force of that state. Starting with the initial wave packet in the ground state and ending with the final state eLMNO/" |0 , the dynamics may be represented by the autocorrelation function PQ *A+ PQ *A+ 0eLMNO/" 0.
(11)
The final state is represented by a linearly displaced harmonic oscillator, keeping the same harmonic frequency as in the ground state. The ground and excited nuclear Hamiltonians (expressed in dimensionless quantities) are ℏω 2 P + q2 ) ( 2
(12)
ℏω 2 P + (q + ∆ )2 ) ( 2
(13)
Hg =
He =
where ∆ is the upper linear dimensionless displacement with respect to the initial state. Introducing the following unitary transformation, eLMNO/" exp R6G*ST U S+A/"V,
(14)
which allows avoiding dealing directly with the excited nuclear Hamiltonian. S in Eq. 14 is the unitary translational operator. Inserting Eq. 14 in Eq. 11 leads to PQ *A+ W
∆
√
ZeLM[O/" Z
∆
\.
√
(15)
Constructing similar PDE to that in Eq. 5 and solving it using the method of characteristics17 leads to ∆2 iω t Fh (t ) = exp − (1 − e − iωt ) − . 2 2
(16)
The result in Eq. 16 is in exact agreement with the autocorrelation function of displaced harmonic oscillator. III.
Morse Oscillator Thermal States and Quantum Dynamics
In this section we will apply the approach of PDE to reproduce the equilibrium canonical partition function of Morse oscillator. We will also apply the same approach to evaluate the 5
optical time autocorrelation function of the same anharmonic oscillator. The Morse oscillator Hamiltonian assumes this form (expressed in dimensionless quantities)
H=
(
ℏω 2 1 P + 1 − e− 2 2χ
2χq
) , 2
*17+
where χ is the anharmonicity constant. A. The equilibrium canonical partition function
While |〉 was used in the previous section to denote harmonic coherent states, |^〉 will be used here to signify Morse coherent states. It is therefore useful to provide a brief account of Morse oscillator coherent states. Morse coherent states may be defined as eigenstates of the lowering operator _ (sometimes referred to as annihilation operator) as _|^〉 z|^〉
*18+
Un-normalized Morse coherent states written as a linear combination of the Morse oscillator eigenstates |J〉as |^〉 ∑f/
bc Dgh de*D+ |J〉
*19+
where ρ(m) will be obtained by using the definition of Klauder-Perelomov coherent states, namely |^〉 49:*^_T +|0〉,
(20)
where _T is the anharmonic raising operator and |0〉 is the vacuum state of Morse oscillator. The quantum number m in Eq. 19 signifies the number of anharmonic phonons with m = 0, 1, …,
[N/2], where j k"$ 2$N 6 1, with o and being the reduced mass and the Morse parameter lmn
which governs the breadth of Morse potential, respectively. Expanding the above exponential in Taylor series and carrying out the _T operation yields11 6
p*J+
qr s*Dt+s*fDt+ s*ft+
(21)
Note that coherent states in Eq. 19 were left un-normalized for convenience and better book keeping. It is also helpful at this point to define the ladder operators for Morse oscillator, _k
"u nN
_T k
v
"u nN
nN
"u
v
nN
"u
6
K
w x
xw
w x
6 z,
xw
w y
6 z, w y
(22)
(23)
where the Morse coordinate { *4S} /"~+4 2 with S} , ~, and being the dissociation energy, harmonic frequency, and the Morse parameter, respectively. Assuming that Morse potential will have [N/2] finite bound states, the inner product of two coherent states |^ 〉 and |^ 〉 is ^ |^ ∑Rf/V Dgh
*b ∗ b$ +c e*D+
*24+
One would need the closure relationship in order to evaluate time correlation functions. One may start with the projection operator (assuming the system of interest escapes dissociation state so as to warrant completeness of the vibrational bound states) ∑Rf/V Dgh |J J |
(25)
which will lead to the following closure relation in terms of Morse coherent states (unnormalized) |^ ^| ,
(26)
where the form of the integration measure dσ that I have used in Eq. 26 is missing the normalization factor since I have chosen my coherent states to be un-normalized,
7
*ft+
$ ||$ f t
^.
(27)
The equilibrium canonical partition function of Morse oscillator may be expanded in terms of its coherent states as6
*ft+ f
b} b
$ ||$ t
^
(28)
where d2z = dxdy, with ^ 9 K G{. One may also express z in polar coordinates as ^ 04 . For the Morse oscillator the exponential Boltzmann operator may be recast as e |^〉 49: 6 vHh K ^
b
6 ^
$
b $
z |^〉,
(29)
where *1 6 +"~ and "~ . Hh is the zero-point energy of the Morse oscillator. The
differential operators ^ b and ^ b $ commute for which the reason they may be utilized interchangeably. Acting successively with these two operators Eq. 29 becomes
$
e |^〉 49:R6Hh V 49: 6^
b
49: ^
$
b $
|^〉.
(30)
Proceeding along the same lines as in Sec. II, two PDEs along with their initial conditions will be obtained. Assuming that our physical function that needs to be operated on with exp *^ b+ is
Ψ(η, z) while for the second order exponential operator exp *^ b $ + it is G*θ, z+, then the firstand second order (parabolic) partial differential equations (PDE) need to be solved are $
and
Ψ*, ^+ ^
b
Ψ*, ^+
(31)
G*, ^+
(32)
Ψ*0, ^+ Φ*^+
G*, ^+ ^
$
b $
G*0, ^+ *^+. 8
Utilizing the method of characteristics, the PDE in Eq. 31 is readily solved to yield Ψ*, ^+ Φ*4 ^+.
(33)
Ψ*, ^+ ∑∞ gh *+^ .
(34)
f/ Ψ*, ^+ h K z K ∑g exp * * 6 1++^ ,
(35)
While the parabolic PDE in Eq. 32 may conventionally be transformed into the heat/diffusion equation using Laplace transform, it was found easier to use the power series method directly
to find
where and are expansion coefficients. Now equating G*0, ^+ to the inner product of Morse coherent states, by taking *^+ ^|^ , will immediately render the final solution, f/ Γ*ft+ b ∗ 49:* !*f+! f
Ψ*, ^+ ∑gh
* 6 1++^ .
Using Eqs. 33−36 leads to
49: 6 vH0 K ^
4 ∑gh
^
6 ^2
f/ Γ*ft+ b ∗ f 49:* !*f+!
2
^2
z *^+
* 6 1++^4
(36)
(37)
Alternatively, one can get a closed-form expression using series expansion, leading to 49: ¡6 ¢Hh K ^ |^| £ ¢1 K j
f
6 ^ £¤ *^+ ^ ^
f⁄
j K |^| 4 2|^| 4 ¢2 K £ ¥ ∗ ¦ j ^ *^4 + 9
f
× U v6 , , yb ∗ $ f
z *b} © +$
*ft|b|$ } © +$
where *^+ ^|^ and °*, ±, ^+ is the confluent hypergeometric function of the second kind.20
*38+
Using the above approach, one can take a closer look at the thermal states of Morse oscillator derived recently by Popov et al15 and obtain a more direct closed-form rather than leaving it in a differential form. The reported form in Ref. 15 reads as 4 c 49:R6Hh V 49: $ 49:R6JV.
(39)
²*C, + 49: $ 49:R6JV.
(40)
$
In this case we have $
Proceeding along the same approach established above, one can construct this PDE and its initial value at c = 0
³
² *C, +
$
$
² *C, +
²*0, + ´*+ 49:R6JV,
(41)
where C . The PDE in Eq. 41 is the heat/diffusion equation, on an infinite line in space,
of which solution is
K*C, +
t¶ { √y³ ¶
4 *w+
$ /y³
49:RJ*C J 6 +V.
49:R6J{V
(42)
The result in Eq. 32 offers numerical efficiency that falls out of this sort of stable, smooth Gaussian integrals, which really applies to thermal states of many important systems. It is noteworthy that the treatment of Popov that led to Eq. 39 had imposed an approximation with respect to the number of bound vibrational states, whereas the results reported in Eqs. 37 and 38 bear no approximations and are exact. B. Dynamics and autocorrelation function for weak electron-phonon coupled systems 10
In this section an expression for the autocorrelation function of Morse oscillator using Morse coherent states will be derived. The general formula of the optical autocorrelation function is given in Eq. 10. Starting with the Morse oscillator vacuum state|0〉, the molecular wavepacket makes a transition to the electronic excited state where it evolves under the nuclear force of excited displaced Morse potential. Starting with the initial wave packet in the ground state and ending with the final state eLMNO/" |0 , looking at the Morse oscillator autocorrelation function P*A+ P*A+ 0eLMNO/" 0
(43)
will impart valuable information on the system dynamics. The final state is represented by a linearly displaced Morse oscillator, keeping the same harmonic frequency as in the ground state. The ground and excited nuclear Hamiltonians are
(
Hg =
ℏω 2 1 P + 1 − e− 2 2χ
He =
ℏω 2 1 P + 1 − e− 2 2χ
2χQ
) 2
(44)
(
2 χ (Q −∆ )
) , 2
where ∆ is the upper linear dimensionless displacement with respect to the initial state. eLMNO/" |0 is very hard to carry out algebraically, therefore one may get around that by introducing the following unitary transformation, eLMNO/" exp R6G*ST U S+A/"V,
(45)
S 49:;6∆*_ 6 _T +/√2<,
(46)
∆/√2eLM[O/" ∆/√2.
(47)
which allows avoiding dealing directly with the excited nuclear Hamiltonian. D is a unitary displacement operator and takes on this form
where _ is Morse annihilation operator and _T isthe corresponding creation operator. Inserting Eq. 46 in Eq. 45 leads to
11
∆/√2 is a Morse coherent state and therefore the above relations used to find the partition function will apply. Equation 48 will be evaluated assuming anharmonic weak electron-phonon coupling (conventionally when Huang-Rhys-factor S ≤ 1 where ¸ ¹ /2), leading to compact expressions.
Using the definition of Morse coherent states in Eq. 19 one can show that eLM[O/" |^〉 49: 6G vHh K ^
where *1 6 +"~ and ". autocorrelation function would read P*A+ W
∆
√
b
6 ^
$
b $
z A/" |^〉,
(48)
After utilizing operator algebra techniques the
Z49: 6G vHh K ^
b
6 ^
$
b $
z A/" Z
∆
\.
√
(49)
Developing analogous PDEs as those in Eqs. 31-33 and 38 yields the autocorrelation function for Morse oscillator in the weak electron-phonon coupling limit
f
P*A+ v1 K fz º$
¢2 K
»¼© " º$
f
f
£ ½
$ »¼© L¢ " ftº$ £ "
¾º¿
f⁄
À
° ¢6 , ,
»¼©
f L* " ftº$ +$ "
y¾º¿
£
*50+
where °*, ±, . + is the confluent hypergeometric function of the second kind.20 Careful examination of Eq.50 reveals that the Franck-Condon factor (FCF) for the 0-0 transition is | 0|0 | v1 K z f º$
f
.
(51)
One may recover the FCF for the 0-0 transition of a displaced harmonic oscillator by taking j → ∞, yielding Ä$
| 0|0 | 4 $ .
12
(51)
Equation 51 is the exact FCF documented in the literature. This end, it would be helpful to compare the time correlation function in Eq. 50 to the exact dynamics6 of a system with ~ 50 cm-1, ∆1, and χ 0.0025. Figure 1 shows this calculation where the two curves of Eq. 50 and that of Ref. 6 are identical. IV. Concluding Remarks
The present approach tends to avoid dealing directly with operator algebra and the formidable algebraic techniques needed to manipulate coherent states. The involved algebra gets even more complicated when Morse oscillator coherent states are utilized.12-14 Because the herein approach relies on partial differential equations (most often parabolic if second order), not only does it encompass a broader range of anharmonic potentials, but also offers high numerical efficiency. One can appreciate this by taking a closer look at Morse oscillator eigenfunctions,21 for example. They seem to have numerical instability and convergence issues, for which the reason utilization of Kratzer oscillator was suggested, and discussed in detail, in case of anharmonicity.22 Although using coherent states solves the problem of dealing directly with eigenstate representation, thereby eliminating numerical instability, it introduces another problem−operator algebra and other insurmountable infinite series issues. This paper suggests a PDE approach for dealing with different oscillator coherent states instead of operator algebra manipulation. Harmonic oscillator partition function and autocorrelation function have been derived to ratify the correctness and applicability of the method. Exact agreement of the results shows the success and utility of the method. The utility of this method, though only applied to harmonic and Morse oscillators here, lies in its Gaussian integrals that result upon solving the respective PDEs, especially when modeling requires more complicated potentials. In this case, those integrals feature smoothness and stability. Thus, our approach of utilizing PDEs in solving for quantum dynamics related problems should set the stage for wider applicability in case of more complicated anharmonic systems, as numerical routines are well established for evaluating integrals or solving PDEs. Closed-form expressions for both the partition function and autocorrelation function of Morse oscillator have been derived. In fact, our PDE approach has enabled us to report two exact analytical expressions bearing no restrictions or approximations, e.g. Eqs. 37 and 38.
References
13
1. Wu, J.; Cao, J., J. Chem. Phys. 2001, 115, 5381. 2. Toutounji, M., J. Phys. Chem. C 2010, 114, 20764. 3. Toutounji, M., J. Phys. Chem. B 2011, 115, 5121. 4. Toutounji, M., Phys. Chem. Chem. Phys. 2012, 14, 626. 5. Strekalov, M. L., Chem. Phys. Lett. 2007, 439, 209. 6. Toutounji, M., Theor Chem. Acc. 2014, 133, 1461. 7. Toutounji, M., Chem. Phys. Lett. 2013, 555, 92. 8. Toutounji, M., J. Chem. Phys., 2008, 128, 164103. 9. Popov, D., Phys. Lett. A 2003, 316, 369; EJTP 2006, 11, 123. 10. Daoud, M.; Popov, D. Int. J. Mod. Phys. B 2004, 18, 325. 11. Popov, D.; Zaharie, I.; Dong, S., Czech. J. Phys. 2006, 56, 157. 12. Nieto, M., M.; Simmons, L., M., Phys. Rev. A1979, 19, 438; ibid, Phys. Rev. D 1979, 20,1321; Phys. Rev. D 1979, 20,1342; ibid, Phys. Rev. D 1980, 22, 403. 13. Jellal, A., Mod. Phys. Lett. A 2002, 17, 671. 14. M. Angelova, A. Hertz, V. Hussin, Journal of Physics A: Math. Theor. 2012, 45, 1751. 15. Popov, D.; Dong, S.; Pop, N.; Sajfert, V.; Simon, S., Annals of Physics, 2013, 339, 122. 16. Castanos, O.; Lemus, R., Mol. Phys. 2010, 108, 597. 17. Arfken, G., Mathematical Methods for Physicists; Academic Press, California, 1985. 18. Castlillo, J., F.;Reid, K., L., Laser Chemistry, 1999, 19, 57. 19. Robinett, R. W., Physics Reports, 2004, 392, 1. 20. Gradsteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series and Products; Academic Press: New York, 2007. 21. Toutounji, M., J. Phys. Chem. B 2011, 115, 5121. 22. Toutounji, M., J. Chem. Theory and Compt. 2011, 7, 1804.
14
Figure Caption
Figure 1: Two dipole moment autocorrelation functions of an anharmonic system with ~ 50 cm-1, ∆1, and χ 0.00325 , both of which coincide and mach exactly. While the ordinate takes arbitrary unites, the abscissa is labeled with femtoseconds (fs).
0
1000
2000 Tim e fs
15
3000