Time-correlation function and average energy of molecules in presence of Deng-Fan potential in a moving boundary

For the Deng-Fan potential within a moving boundary condition, the time-dependent Schrödinger equation is considered analytically. The eigenvalue equation is solved by using a combination of Pekeris and Greene-Aldrich approximations. Various time-dependent quantities including density distribution function, auto-correlation function, disequilibrium, average energy, quantum similarity, and quantum similarity index are obtained for selected eight diatomic molecules. The motion of the peak of the density function, with moving boundary condition is investigated for ground states of some diatomic molecules along with the corresponding peak values.

atomic distance) is essential. Ever since the publication of first three-parameter, analytical potential [1], perhaps the oldest one, a plethora of attempts have been made to design progressively better functions. The correct knowledge about their eigenvalues and eigenfunctions leads to valuable information regarding transition frequency and matrix elements. An ideal potential function should satisfy the following limiting conditions at small and large distances, namely, V (0) = ∞ and remains a constant as r → ∞. While Morse potential suffers from certain limitations (e.g., remains finite at r → 0) and thus exhibits considerable disagreements with experimental observations, it does provide a basis for construction of improved functions. This has remained an active area of research covering a large span of time, with widespread applications in molecular spectroscopy atom/molecule adsorption on solid surface, deformation of cubic metal, etc. Some important and popular models for vibrational interactions in molecules are as follows: Manning-Rosen [2][3][4][5], Húlthen [6][7][8][9], Woods-Saxon [10,11], Pöschl-Teller [5,12,13], Tietz-Hua [14][15][16], pseudoharmonic [17][18][19], Rosen-Morse [20][21][22], Kratzer [23,24], Eckart [25] and so on.
In the current work, we focus on yet another potential function, namely, Deng-Fan (DF) potential for diatomic molecules, which was introduced about 70 years ago. Being a three-parameter function, it can be expressed as below where the (+)ve parameters D e , r e , α refer to dissociation energy, equilibrium internuclear distance and potential well radius. In the limit of internuclear distance approaching zero and infinity, it provides a qualitatively correct behaviour. Due to its close resemblance to Morse potential, it is often referred as Generalized Morse potential [26,27]. Its connection to another important diatomic potential (Manning-Rosen) has been discussed as well. A shifted (by dissociation energy) sDF potential [28] has generated much interest in the literature in recent years. For large r region (r ≈ r e , r > r e ), it closely resembles the Morse potential, but differs at r ≈ 0. Also, the deep (D e 1), DF and sDF potentials can be approximated by harmonic oscillator in r ≈ r e region [26].
For an arbitrary angular momentum state, the eigenvalues and eigenfunctions cannot written down in closed analytical form exactly. Thus various approximations were reported in relativistic and non-relativistic domain; the literature is quite vast. We mention here some prominent ones. The exact solvability problem has been addressed in [26] via an SO(2, 2) symmetry algebra. The circular, nodeless states were studied by algebraic method [29]. Approximate analytical eigensolutions for rotating DF potential were expressed in terms of generalized hyper-geometric functions 2 F 1 (a, b; c; z) [30] in a given quantum state. Within a super-symmetric shape invariance method, an improved approximation to the centrifugal term was adopted in [31]. An analytical solution for Dirac equation has been reported within the above method [32]. In [33], solutions of Klein-Gordon equation for nonzero-states were discussed. In another attempt, a Pekeris approximation for centrifugal term within a Nikiforov-Uvarov (NU) framework was advocated for Dirac, Klein-Gordon as well as Schrödinger equation (SE) [28,34,35]. The Klein-Gordon equation in Ddimension was treated by means of an ansatz method in bound states [36], while scattering states were reported in [37]. Accurate results were reported by means of numerical generalized pseudospectral [38] method. Apart from these, one can also find exact quantization rule [39], as well as a Feynman path-integral formalism [40] coupled with an approximation for centrifugal term, for the generalized D-dimensional problem.
Expectation values [39] and thermodynamic properties [41,42] were also considered. For a critical and comparative analysis of the performance of Morse, Manning-Rosen, Schlöberg and DF potential in the context of diatomic molecules, one may consult [43].
In this communication, at first, we find exact solution of time-dependent (TD) SE in presence of DF potential with moving boundary condition. For quantum systems, time-dependent Schrödinger equation is of great general interest in the linear as well as nonlinear domain [44][45][46]. In this article, we have considered the latter with moving boundary condition. Then we obtain analytical expressions of certain expectation values and time-dependent average energy. Then we pursue a host of time-dependent quantities including time-correlation function, quantum similarity between two states and disequilibrium of a given state. For illustration, we report numerical values of these quantities in case of eight representative diatomic molecules, viz., H 2 , LiH, HCl, ScH, TiH, CrH, VH and CO. Moreover, we also compute the quantum similarity index (QSI) between two diatomic molecules. In Sect. 2, we describe the theoretical methodology of the timedependent problem in quantum system with moving boundary condition. In Sect. 3, we make a detailed presentation of the results along with a discussion in the context of these molecules. Finally, we conclude with a few remarks in Sect. 4.

Theoretical methodology
In this article, we are interested in the time-dependent Schrödinger equation (TDSE), as below where the symbols have their usual meanings. The particular TD potential used here has following form [47,48]: where x 0 and t 0 are scale factors of x and t such that X = x/x 0 , T = t/t 0 ; c signifies speed of light andh is the reduced Planck constant. The quantities x 0 , t 0 have dimensions of length and time, such that ct 0 has a dimension of length. The functions β and L are dimensionless scale functions of T , while the function x 0 β(T ) represents the moving boundary condition of solutions of TDSE (2). Equation (2) is solvable in presence of the TD potential, Eq. (3) by the method of invariant, provided L(T ), β(T ), F 1 (T ) and F 2 (T ) satisfy the Ermakov equation [47][48][49]. Then the corresponding TD quantum system will be adiabatic.
Here, we have chosen the effective potential V e f f to be of DF type, defined in Eq. (1) and a transformation r = (x − x 0 β(T ))/L(T ). Then the corresponding time-independent (TI)SE can be recast in the following form: and the TDSE, Eq. (2) becomes solvable, when the following relations [49,50] hold where (i) is the separation constant, which is equal to the eigen spectrum of the TISE. The variables x and r are defnied as the TD and time-independent position variables, respectively. Naturally, the corresponding TD potential and the time-independent effective potentials are connected via Eq. (3). It has been shown that the exact solution of Schrödinger equation, under such a condition, can be defined by point transformation and separation of variables [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68]. Note that the "effective" potential includes the centrifugal term, Then the corresponding TD potential is given as It may be noted that DF potential is one of the many molecular potentials reported in the literature, over the past several decades. And its merits and demerits in the context of molecular spectroscopy have been well documented in the references [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. The three paramaters present in the potential represent a given molecular system. The particular moving boundary condition can be applicable to other potentials as well; for example, recently it has been applied to Pöschl-Teller potential [50]. It is well known that the DF potential satisfies [69] the conditions given in [70], namely where ω e denotes harmonic vibrational frequency. From Eq. (8), one can find the value of α as below, where P L [z] is the Product log function [71], and it is the principal solution for w in z = we w . Then the TISE is 123 then TD Schrödinger equation can be solved (in order to get the TD average energy, average force and position expectation, etc.) in presence of the potential defined in Eq. (3), or equivalently in Eq. (7). The solutions of Eq. (10) are obtained within a recently proposed scheme combining both Pekeris [72] and Greene-Aldrich [73] approximations. This has lead to a very satisfactory treatment of potential throughout the whole domain of consideration [5,25,42,74,75]. Therefore, solution of Eq. (10) can be expressed as [42], and energy given by, where is the Pochhammer symbol, (a) = is the Gamma function, and λ is a real number lying in between 0 ≤ λ ≤ 1. Accordingly, the normalized wave function of time-dependent potential (7) is derived as (17) Here, a, b have dimensions of length −2 and length −1 , respectively, while c 1 is a dimensionless quantity. The energy of the DF potential can be written as a power series of (n + 1 2 ) in the form where the coefficients γ ,k (k = 0, 1, 2, 3, 4) are defined by In absence of the centrifugal term, the coefficients γ 0,k → 0 as ξ → ∞ for k ≥ 3 which coincides with the reference [26] and the corresponding energy is apprpximated to the Morse-oscillator. The DF potential is very close to the Morse potential in the regions r ≈ r e and r > r e . We also checked that γ 0,k → 0 as D e , ξ → ∞ for k ≥ 3 and then the DF and Morse potentials are approximated in the region r ≈ r e by a shifted harmonic oscillator with frequecy α 2h2 γ 0,1 2μ .

Expectation values and average energy
The expectation of x k with respect to a normalized time-dependent state ψ(x, t) is defined by where r k (i) = ∞ 0 r k |Q(r )| 2 dr (k = 1, 2) are expectation values of r k with respect to corresponding TI state Q. Then standard deviation of x is defined by The average energy of a normalized time-dependent state ψ(x, t) is found to be [49,50,[76][77][78][79][80], Then the expectation of x and x 2 with respect to the state ψ n, (x, t) is defined by where The average energy of time-dependent potential (7) can be expressed as where is the Bell polynomial, where π(m, j) denotes the set of partitions, such that j 1 + j 2 + · · · + j m− j+1 = j, j 1 + 2 j 2 + · · · + (m − j + 1) j m− j+1 = m, and c . In particular, for L = κ, a constant, one obtains

Time-correlation function, disequilibrium and quantum similarity
The time-correlation function between states ψ n 1 , 1 and ψ n 2 , 2 can be defined as [80][81][82], where (x 0 β 12 , ∞) is the common domain of overlap integral of moving boundary solutions, and t 2 ) is called the auto-correlation function of ψ n, at different time domains t = t 1 and t = t 2 but at the same time C (d) (n, ) (t 1 , t 1 ) = 1. The quantum similarity measure between two states with densities ρ n 1 , 1 (x, t) and ρ n 2 , 2 (x, t) can be defined as 123 On the other hand, D The quantum similarity measure (QSM), M (n 1 , 1 ),(n 2 , 2 ) (t), is then written as [50], M (d)

Results and discussions
The molecular parameters D e , r e and μ are taken from ref. [39]. They are listed in Table 1 sin 2 (2π T /3), while for the latter one, we employ L(T ) = 1, β(T ) = 0.5T 2 , respectively. In both cases, x 0 = 1Å. It may be mentioned that the density plots in Figs. 1 and 2 give the resemblance of soliton solitary waves and each density has a peak. The peak point, x = x 0, , of the density function, |ψ 0, (x, t)| 2 , is obtained from the conditions, , as well as the average position, x n, = x 0 β(T ) + L(T ) r n, , of density func-tion are qualitatively similar, but there are numerical differences.
In Table 2 now, we have produced the calculated expectation values of r and r 2 of all the diatomic molecules, in TI quantum system in Å and Å 2 units, respectively, for n = 0, 1, 2, = 0, 1, 2, 3. These cannot be directly compared with reference results, as the literature values are absent. However, it is worth mentioning that the calculated eigenvalues have been checked and compared with other existing results earlier, which shows excellent agreement. These comparisons have been presented earlier [42] to demonstrate the success and efficacy of present method, and hence not repeated here.
It is worth mentioning that in absence of the moving boundary, the DF potential generally offers bound states with real energies, called ro-vibrational energies. There is a maximum possible quantum number (or n) for a fixed n (or ) for which energies are non-negative. But in presence of the moving boundary condition, the average energies are complex, in general. For a particular scaling factor L =constant, we obtain real average energies for a given quantum number n and . Now, one can find a maximum possible quantum number (or n) for a fixed n (or ) for which energies are real and non-negative. In Fig. 4, we have plotted the average energy of eight diatomic molecules in eV units, along two different curves. In panels (A), (B), we have considered periodic moving boundary condition and in (C), (D) parabolic moving boundary condition have been used. The panels (A), (B), (C), (D) provide average energy variations of H 2 , LiH, HCl, ScH, TiH, VH and CrH, CO molecules, respectively. Note that (A), (C) offers these plots for ground state, while the other two panels (B), (D) correspond to states having radial and angular quantum numbers n r = 0 and = 2. The similarity between (A), (B) and (C), (D) is quite visible. The average energies of H 2 and LiH at t = 0 are: 0.3895eV and 0.0742eV, respectively, but their ro-vibrational energies in absence of moving boundary condition are: 0.349980221eV and 0.103334650eV [42] for n r = = 0. Under the choice of scale factor L and moving boundary condition, ψ(x 0 β(T ), t) = 0, one can find average energy, and the corresponding average force acting on the boundary wall in a confined region. This method can be extended to time-dependent confined quantum systems as well.
Now the time-correlation function of two molecules is plotted in Fig. 5. In left and middle columns, we have Table 3 Quantum similarity index of molecules of Table 1  plotted the real and imaginary parts of time-correlation C 0,1 (t, 0). In the last column, we have shown the same in the complex plane of Re C 0,1 (t, 0) , I m C 0,1 (t, 0)] . In the first row, we have compared the timecorrelation function of H 2 with LiH, HCl, ScH, TiH, VH, CrH and CO, which are represented by red, blue, black, green, magenta, orange and cyan colours, respectively. The second row compares the same for LiH with HCl, ScH, TiH, VH, CrH and CO, which is represented by blue, black, green, magenta, orange and cyan lines, respectively. Similarly the last row compares this between CrH and CO. In this row, we have plotted 10 6 C 0,1 (t, 0). From this figure, it is observed that the time-correlation of CO with other molecules is very small compared with other pairs.
As we stated earlier, quantum similarity is time dependent but quantum similarity index is time independent for a given potential [50]. We can see that the potential, Eq. (1) is dependent on parameters, D e , r e , α, μ. Therefore for two sets of parameters the potential has two values. Therefore, for two diatomic molecules, we observe that M (d) (n 1 , 1 ),(n 2 , 2 ) (t) = M (i) (n 1 , 1 ),(n 2 , 2 ) /L(T ). In this case, we follow the definition of Eq. (30) for quantum similarity, and for quantum similarity index, we follow the definition of Eq. (32). In Fig. 6, we plot the quantum disequilibrium of molecules with respect to T . The disequilibrium decreases as T increases, and their qualitative features are similar even though their magnitude is different. The black, blue and red colours represent (i) β 1 (T ) = sin(2π * T ) 2 , L 1 (T ) = √ 1 − 0.25T + 0.25T 2 ; (ii) , having x 0 = 0.01 Å, ct 0 = 250000 Å for the state characterized by quantum numbers, n = 0, = 1. If the scale factor L is constant, then disequilibrium remains constant. However, disequilibrium decreases (or increases) if L(T ) increases (or decreases).
In Table 3, the QSI between two diatomic molecules at t = 0 are given. Next, in Fig. 7, we plot the timedependent QSI between two diatomic molecules. The interval (a i j , b i j ) represents the range of QSI of ith and jth molecules, where i = 1, 2, 3, · · · , 7 represents H 2 , LiH, HCl, ScH, TiH, VH, CrH and j = 1, 2, · · · , 7 refers to LiH, HCl, ScH, TiH, VH, CrH, CO. Table 4 produces the calculated numerical values of a i j and b i j . Note that close similarity of ScH and TiH as evidenced by the QSI values is reminiscent of our previous study [85] of these molecules in presence of a pseudo-harmonic potential. There, we investigated QSI among 19 molecules in presence of pseudo-harmonic and its iso-spectral potentials. Here also, it is found that: (i) QSI between ScH and TiH is maxumum, (ii) CO molecule shows maximum dissimilarity with other molecules, such as, H 2 , LiH, HCl, ScH, TiH, VH, CrH, which is amply clear from Table 3, (iii) four species, namely ScH, TiH, VH and CrH, are quite similar (iv) The QSI between H 2 and scH is minimum which is in conformity with our previous study [85]. The timedependent QSI satisfies all these properties (i)-(iv) in a time-dependent quantum system with moving boundary condition. The time-dependent QSI between two molecules changes with respect to time in a very short range, which confirms that QSI is a fixed number up to a certain round of approximation [50] and it lies between 0 to 1.

Conclusion
Exact solutions of time-dependent Schrödinger equation is obtained in presence of Deng-Fan potential using a simple novel approximation of the centrifugal term. The time-dependent average energy is defined for an arbitrary moving boundary condition and it is expressed in terms of scale factors x 0 , t 0 and L. These are performed for eight representative molecules, namely, H 2 , LiH, HCl, ScH, TiH, VH, CrH, CO. The boundary moves along two different curves. The motion of the peak of density function has been shown anayltically for arbitrary moving moving boundary condition for ground states of some diatomic molecules; the corresponing peak values are defined analytically.
The numerical peak values are defined for parabolic and periodic moving boundary condition. The numerical values of time-dependent QSI between pairs of diatomic molecules are presented in detail. The densities of solitary wave functions of ScH, TiH, VH, CrH molecules are quite similar. Moreover, they are similar with respect to time-dependent average position, average energy, disequilibrium, quantum similarity as well as QSI.