State-Dependent Molecular Dynamics

Abstract

This paper proposes a new mixed quantum mechanics (QM)—molecular mechanics (MM) approach, where MM is replaced by quantum Hamilton mechanics (QHM), which inherits the modeling capability of MM, while preserving the state-dependent nature of QM. QHM, a single mechanics playing the roles of QM and MM simultaneously, will be employed here to derive the three-dimensional quantum dynamics of diatomic molecules. The resulting state-dependent molecular dynamics including vibration, rotation and spin are shown to completely agree with the QM description and well match the experimental vibration-rotation spectrum. QHM can be incorporated into the framework of a mixed quantum-classical Bohmian method to enable a trajectory interpretation of orbital-spin interaction and spin entanglement in molecular dynamics.

2. Mixed Quantum-Classical Mechanics

In this section, we review the basics of the MQCB method by applying it to simulate the motion of a diatomic molecule, which is composed of Nuclei A and B, of mass M_A and M_B together with a number N of electrons (see Figure 1). The internuclear position vector will be denoted by X and the position vectors of the electrons with respect to O, the center of mass of A and B, by r₁, r₂,⋯, r_N. The position vectors of A and B are denoted by R_A and R_B, so that = R_A − R_B. The Schrödinger equation describing the motion of the molecule takes the following form [12,13]:

$$iħ\frac{\partial }{\partial t}\Psi (t,x,X)=\left[-\frac{{ħ}^2}{2m}\frac{{\partial }^2}{\partial x^2}-\frac{{ħ}^2}{2M}\frac{{\partial }^2}{\partial X^2}+V(x,X)\right]\Psi (t,x,X)$$

(2.1)

where x = [r₁ r₂ ⋯ r₃] is the collective vector representing the quantum motion of the electrons and X is the vector representing the classical motion of the nuclei.

Writing the wave function as Ψ(t,x,X) = R_B(t,x,X) exp (iS_B(t,x,X)) with R_B and S_B being real, Equation (2.1) can be separated into three coupled equations in terms of R_B, S_B and their derivatives. For a computed S_B, the Bohmian velocity is given by:

$$\frac{dx}{dt}=\frac{1}{m}\frac{\partial S_B(t,x,X)}{\partial x},\frac{dX}{dt}=\frac{1}{M}\frac{\partial S_B(t,x,X)}{\partial X}$$

(2.2)

where m is the mass of electron and M = M_AM_B/(M_A + M_B) is the reduced mass of the molecule. With the introduction of the quantum potential:

$$Q(t,x,X)=-\frac{{ħ}^2}{2m}\frac{1}{R_B}\frac{{\partial }^2{R}_B}{\partial x^2}-\frac{{ħ}^2}{2M}\frac{1}{R_B}\frac{{\partial }^2{R}_B}{\partial X^2}$$

(2.3)

Equation (2.2) can be recast into classical-like equations of motion:

$$\frac{d^{2}x}{d{t}^2}=-\frac{1}{m}\frac{\partial \left(V(t,x,X)+Q(t,x,X)\right)}{dx}$$

(2.4)

$$\frac{d^{2}X}{d{t}^2}=-\frac{1}{M}\frac{\partial \left(V(t,x,X)+Q(t,x,X)\right)}{dX}$$

(2.5)

A workable scheme of MQCB has been implemented successfully by replacing the full-dimensional wave function Ψ(t,x,X) by a wave function in the x quantum subspace that depends parametrically on the classical position X(t). The approximate wave function obeys the Schrödinger equation:

$$iħ\frac{d}{dt}\tilde{\Psi }(t,x,X(t))=\left[-\frac{{ħ}^2}{2m}\frac{{\partial }^2}{\partial x^2}+V(x,X(t))\right]\tilde{\Psi }(t,x,X(t))$$

(2.6)

where d / dt denotes the material derivative ∂ / ∂t + Ẋ ∙ 𝛻 along the classical trajectory X(t). The MQCB method and the mean-field method share the same quantum degree of freedom described by Equation (2.4), but differ from each other in the way that the classical degree of freedom is governed by the quantum wave function. Instead of Equation (2.5), the classical trajectory X(t) for the mean-field method is computed by the mean potential:

$$\frac{d^{2}X}{d{t}^2}=-\frac{1}{M}\frac{\partial }{dX}{V}_{\text{eff}}(t,X)=-\frac{1}{M}\frac{\partial }{dX}{\displaystyle \int {\left|\Psi (t,x,X)\right|}^2}V(x,X)dx$$

(2.7)

Upon applying Equation (2.5) or Equation (2.7) to a diatomic molecule, we find that it is convenient to express the vector X in the spherical coordinates (r,θ,ϕ) in order to describe the vibration and rotation of the two atoms with respect to their center of mass. However, the expressions of quantum motion described by Equation (2.2) to Equation (2.7) are valid only for Cartesian coordinates. To give an explicit manifestation of the orbital and spin angular motion of the molecules, the reformulation of Bohmian mechanics under spherical coordinates is necessary.

It will be shown in the next section that QHM provides MQCB with state-dependent molecular dynamics, which preserve the quantization property of orbital and spin angular momentum. By contrast, in the mean-field method, the classical trajectories X(t) over different quantum states are averaged out to give a state-independent result, as can be seen from Equation (2.7). To compare with the QHM formulation discussed later, here let us have a quick look on the description of the vibration-rotation motion of a diatomic molecule by the classical Equation (2.7). With the effective potential V_eff(t,X) = V_eff(t,r,θ,ϕ) given by Equation (2.7), the classical Hamiltonian governing the 3D relative motion of the two nuclei with respect to their center of mass can be expressed by:

(2.8)

The molecular dynamics (r(t),θ(τ),ϕ(t)) can be solved from the Hamilton equations:

(2.9)

(2.10a)

(2.10b)

(2.10c)

It is clear that the vibrational and rotational motions of the molecule are determined uniquely by the effective potential V_eff(t,r,θ,ϕ) with initial conditions (r(0),θ(0),ϕ(0)) and (p_r(0),p_θ(0),p_ϕ(0)). In the case of a central-force potential V_eff(r), the total energy H_c, the squared angular momentum L² and the z-component angular momentum L_z are conserved quantities along any dynamic trajectory (r(t),θ(τ),ϕ(t)):

(2.11)

where the three conservation constants depend continuously on the initial conditions. In the next section, a quantum version of the classical Hamilton Equations (2.9) and (2.10) will be derived by QHM. The resulting quantum Hamilton equations can describe molecular dynamics in a general non-stationary quantum state Ψ(t,r,θ,ϕ), and a quantum version of the conservation law (2.11) comes out naturally, if Ψ(t,r,θ,ϕ) is an eigenfunction.

3. Quantum Hamilton Mechanics

Consider the diatomic molecule as shown in Figure 1b. Let q be the internuclear position vector expressed in a curvilinear coordinate system, V_eff(t,q) be the instantaneous internuclear potential, and M be the reduced mass. The Schrödinger equation describing the molecular motion can be recast into the quantum Hamilton-Jacobi equation [17,20]:

$$\frac{\partial S}{\partial t}+{H_{\Psi }(t,q,p)|}_{p=\nabla S}=\frac{\partial S}{\partial t}+{\left[\frac{1}{2M}p\cdot p+V_{\text{eff}}(t,q)-\frac{\text{i}ħ}{2M}\nabla \cdot p\right]}_{p=\partial S/\partial q}=0$$

(3.1)

where the action function S is the logarithmic wave function defined as:

S(t,q) = −iħln Ψ(t,q) .

(3.2)

The canonical momentum p in Equation (3.1) is related to the action function S via the law of canonical transformation:

(3.3)

The appearance of the imaginary number indicates that the momentum p has to be defined in a complex domain. In Cartesian coordinates, we have and the particle’s velocity can be determined by the wave function Ψ as

(3.4)

which is the governing equation in the complex-valued Bohmian mechanics [21]. It is the complex momentum p from Equation (3.3) rather than the real momentum from Equation (2.2) that matches the momentum distribution provided by standard quantum mechanics [20]. The relationship between the real and complex momentum can be found as:

$$\frac{\partial S}{\partial q}=\frac{\partial S_B}{\partial q}-iħ\frac{1}{R_B}\frac{\partial R_B}{\partial q}$$

(3.5)

It can be seen that the real Bohmian trajectory solved from only carry information about the dynamics of the momentum flow, while the complex trajectory solved from also includes information about the probability R_B = |Ψ|. The dynamics in the complex phase space (q,p) thus explains in a natural way how to get the correct momentum distribution and explains why algorithms based on complex trajectories are stable and accurate [22,23,24,25,26,27,28]. The advantages of implementing numerical codes in a complex phase space are similar to those of considering complex fields instead of real ones in electromagnetism [20].

When compared with the classical Hamiltonian H_c in Equation (2.8), the quantum Hamiltonian H_Ψ defined in Equation (3.1) has an additional term called complex quantum potential, which in the state Ψ can be expressed by:

(3.6)

The state-dependent molecular dynamics to be developed here all originate from the state-dependent nature of Q. With the quantum Hamiltonian H_Ψ defined in Equation (3.1), the accompanying Hamilton equations appear as the usual form:

(3.7)

The Hamilton equations are coordinate-independent and valid for arbitrary coordinate systems. Especially, in the Cartesian coordinates, we have ∂H_Ψ/∂q = p/M and the first set of the Hamilton equations (3.7) turns out to be the Bohmian velocity, like the one defined in Equation (2.2).

We now specialize q to the spherical coordinates q = (r,θ,ϕ) with r denoting the internuclear distance, and (θ,ϕ) denoting the orientation of the molecule. By expanding the inner product p ∙ p and the divergence ∇∙ p in spherical coordinates, quantum Hamiltonian H_Ψ in Equation (3.1) can be expressed by [17]:

(3.8)

This is the quantum counterpart of the classical Hamiltonian defined in Equation (2.8). Substituting H_Ψ into Equation (3.7), we obtain the first set of the Hamilton equations as:

(3.9a)

(3.9b)

(3.9c)

where the canonical momentum (p_r, p_θ, p_ϕ) has been given according to Equation (3.3):

(3.10)

Equation (3.9) is the quantum counterpart of the classical equation (2.9) and provides the vibrational and rotational dynamics (r(t), θ(τ), ϕ(t)) of the molecule in the state Ψ(t,r,θ,ϕ), which may be stationary or non-stationary. On the other hand, it can be shown that the second set of Hamilton equation (3.7) with p given by Equation (3.3) is just the Schrödinger equation. In other words, the two sets of Hamilton equations (3.7) provide a new approach to MM/QM formulation, where the first set plays the role of MM to derive the molecular dynamics (3.9), while the second set plays the role of QM to give the time evolution of the wave function Ψ.

The quantum Hamiltonian (3.8) can be expressed in a more comprehensive way with the help of Equation (3.9):

(3.11)

where the first three terms constitute the classical kinetic energy T, and the last two terms form the total potential:

(3.12)

The dynamic equations of motion for the molecule can be described either by the Hamilton equations (3.9) or by the Lagrange equations based on the quantum Lagrangian:

(3.13)

from which the quantum Lagrange equations of motion can be obtained as

(3.14a)

(3.14b)

(3.14c)

It can be seen that apart from the internuclear force produced by V_eff, there are additional quantum forces acting on the molecule yielded by the quantum potential Q. In the next two sections, we proceed to show that the action of the quantum potential leads to the state-dependent molecular dynamics compatible with the description of the wave function Ψ.

4. State-Dependent Molecular Vibration

By replacing Equations (2.2) and (2.5) with Equations (3.9) and (3.14), respectively, the computational procedures of MQCB developed in the literature can be used to simulate molecular dynamics including vibration, rotation and spin motions. Before this new QM/MM approach becomes workable, we have to verify that the governing equations derived in Section 3 yield correct molecular dynamics. The verification is based on the comparison with the quantum mechanical description of a diatomic molecule for which the Schrödinger equation relating to the nuclear motion has an analytical solution. The test model is illustrated in Figure 1a, which is an equivalent single-particle model of a diatomic molecule with rotational and vibrational motion described by the spherical coordinates. The effective internuclear potential is modeled by the Morse function and the corresponding Schrödinger equation is solved analytical in the appendix with eigenfunction Ψ_n,J,mJ (r,θ,ϕ) = R_n,J (r)Θ_J,mJ (θ)Φ_mJ (ϕ) given by Equation (A15) and eigenvalue E_n,J given by Equation (A17). All of the parameters and constants appearing below refer to the Appendix.

Substituting Ψ_n,J,mJ (r,θ,ϕ) into Equation (3.9), we obtain the state-dependent molecular dynamics as:

(4.1a)

(4.1b)

(4.1c)

where r = βr is the dimensionless radial distance and τ = t(ℏβ²/ M is the dimensionless time. In a quantum state specified by the three quantum numbers n, J and m_J, the internuclear distance r(t) and the molecular orientations θ(τ) and ϕ(t) can be expressed as functions of time by solving Equation (4.1).

As a comparison, if the standard Bohmian mechanics based on Equation (2.2) is used to simulate the molecular dynamics in the eigenstate Ψ_n,J,mJ (r,θ,ϕ), we will find the molecule to be motionless in all the states with J = 0. The constitution of the complex momentum in Equation (3.5) explains why it is possible to observe non-vanishing momentum in cases where the Bohmian momentum ∇S_B vanishes.

The first test on the accuracy of the state-dependent molecular dynamics is to examine the quantization laws existing in the eigenstate Ψ_n,J,mJ (r,θ,ϕ). In the previous section, we have seen that in the conventional MM description, the three conservation constants depend continuously on the initial conditions, being unable to manifest the quantization laws. Now replacing Equations (2.8) and (2.9) by Equations (3.8) and (3.9), we can derive the expected quantization laws. With the eigenfunction Ψ_n,J,mJ (r,θ,ϕ) given by Equation (A15) and (p_r,p_θ,p_ϕ) given by Equation (3.10), the Hamiltonian H, the squared angular momentum L² and the z-component angular momentum L_z defined in Equation (3.8) can be computed as:

(4.2a)

(4.2b)

$$H(r(t),\theta (t),\varphi (t))=\frac{1}{2M}\left[p_r^2+\frac{ħ}{\text{i}}\left(\frac{2}{r}{p}_r+\frac{ħ}{\text{i}}\frac{{\partial }^2\ln \Psi (r,\theta ,\varphi )}{\partial r^2}\right)\right]+\frac{L^2}{2M{r}^2}+V_{\text{eff}}(r)=E_{n,J}$$

(4.2c)

We find that the resulting values of , L² and H are independent of time and are quantized according to the three quantum numbers n, J and m_J. Unlike the probabilistic interpretation in standard quantum mechanics, here, we have given a dynamic interpretation of the quantization laws by showing that the three physical quantities have constant discrete values along any quantum trajectory (r(t),θ(τ),ϕ(t)) solved from Equation (4.1) in the quantum state specified by the three quantum numbers (n,J,m_J).

The second test is on the consistence between the predictions of the equilibrium bond length made between quantum mechanics and state-dependent molecular dynamics. We will show that the equilibrium bond length r_eq that maximizes the radial probability P_r satisfies the dynamical equilibrium condition dr / dt = 0. We first prove this property for the ground vibrational state. Referring to the Appendix, the corresponding radial wave function with n = 0 is given by

R_0,J (r) = r⁻¹e^−z/2z^α/2, z = 2ηe^{−β(r − r₀)} .

(4.3)

The dependence of R_0,J (r) on the angular quantum number J is reflected in the relation of α and η to J, as shown in Equation (A14). Substituting R_0,J (r) into Equation (4.1a) yields the equation of motion for the rotation-dependent vibration in the ground state:

(4.4)

The above radial dynamics has a stable equilibrium point at z_eq = α, i.e.,

r_eq = b − ln(α / 2η).

(4.5)

The other equilibrium point is at z_eq = 0, i.e., r_eq = ∞, corresponding to the condition of molecular dissociation. Equation (4.5) expresses the equilibrium bond length r_eq as an explicit function of the angular momentum quantum number J. This closed-form expression describes analytically how the equilibrium bond length r_eq increases monotonically with the angular quantum number J. Of significance is that the equilibrium bond length r_eq obtained from the molecular dynamics (4.1a) always coincides with that obtained by QM method. This coincidence has its theoretic origin. We recall the definition of the radial probability:

(4.6)

by noting that R_n,J (r) given by Equation (A13) is a real function of r. On the other hand, according to the dynamic Equation (4.1a), the equilibrium position r_eq satisfies the condition:

(4.7)

It appears that the equilibrium condition (4.7) is just the condition requiring the radial probability P_n,J in Equation (4.6) to have an extreme value.

All of the properties obtained from the dynamics equation (4.1a) can be re-derived from the action of the radial total force . As an illustration, we consider the case with J = 0 for which Θ_J,mJ (θ) = 1 and the total potential defined in Equation (3.12) has a simple expression,

(4.8)

where we note α = 2λ − 1 and η = λ from Equation (A14) for the case of J = 0. The radial total force now can be determined from as

(4.9)

The internuclear distance free from radial force can be found from the condition f_r (r) = 0, which leads to

(4.10)

This value is exactly the equilibrium bond length r_eq already obtained in Equation (4.5) by using the equilibrium condition dr / dt = 0.

The evaluation of in the vicinity of r_eq gives , if r > r_eq, and , if r < r_eq. The sign of in the neighborhood of r_eq indicates that the two nuclei attract each other when their distance is longer than r_eq and repel each other when their distance is shorter than r_eq. Consequently, there is always a restoring force to make r return to its equilibrium position r_eq.

Figure 2 gives a numerical verification of the coincidence of four positions for the state with n = 1 and J = 0: (a) the local minimum of the total potential; (b) the position with f_Total (r) = 0; (c) the equilibrium position with zero velocity dr / dt = 0; and (d) the local maximum of the radial probability P_1,0, where the first three positions come from the state-dependent molecular dynamics, while the last position from the QM description.

Figure 2. Illustration of the coincidence of four positions for the state with n = 1 and J = 0: (a) The local minimum of the total potential V_Total and the local maximum of the radial probability P_1,0; (b) The position free from the radial force f_Total (r) = 0; (c) The equilibrium position with zero velocity dr / dτ = 0.

Figure 2. Illustration of the coincidence of four positions for the state with n = 1 and J = 0: (a) The local minimum of the total potential V_Total and the local maximum of the radial probability P_1,0; (b) The position free from the radial force f_Total (r) = 0; (c) The equilibrium position with zero velocity dr / dτ = 0.

The complex trajectories of r(τ) solved from Equation (4.4) for the four types of diatomic molecules are shown in Figure 3a. It can be seen that the complex trajectories are closed contours circulating about their respective equilibrium positions r_eq computed from Equation (4.5) by using the molecular parameters listed in Table 1. We find that the period of vibration is independent of the actual trajectories and is quantized with respective to the angular quantum number J. This trajectory-independent property can be proven by applying the residue theorem to Equation (4.4):

(4.11)

where the contour c is an arbitrary closed trajectory solved from Equation (4.4). The contour integral depends only on the poles enclosed by the contours. Because all of the trajectories enclose only one pole at z = α, the contour integral has only one possible value equal to the residue evaluated at this pole. Due to the dependence of α on the angular quantum number J as shown in Equation (A14), the resulting period of vibration is allowed only for some discrete values determined by J.

Figure 3. (a) The ground-state quantum trajectories solved from Equation (4.4) for the four types of diatomic molecules on the complex plane of r. The complex trajectories are closed contours circulating around their respective equilibrium positions r_eq; (b) The time responses of Re(r(t)) give the periods of vibration as T_H2 = 0.3840, T_HCl = 1.2496, T_O2 = 0.1203, and T_N2 = 0.0941, from which the vibrational frequencies f = 1/T = ℏβ² / (MT) are computed and compared with the experimental data listed in Table 1.

Figure 3. (a) The ground-state quantum trajectories solved from Equation (4.4) for the four types of diatomic molecules on the complex plane of r. The complex trajectories are closed contours circulating around their respective equilibrium positions r_eq; (b) The time responses of Re(r(t)) give the periods of vibration as T_H2 = 0.3840, T_HCl = 1.2496, T_O2 = 0.1203, and T_N2 = 0.0941, from which the vibrational frequencies f = 1/T = ℏβ² / (MT) are computed and compared with the experimental data listed in Table 1.

Figure 3b illustrates the time responses Re(r(τ)) for the four diatomic molecules in the ground state n = J = 0, wherein the periods of vibration have a simple expression

(4.12)

The λ values of the four molecules and their periods of vibration computed by Equation (4.12) are listed in Table 1. Also shown in Table 1 are the experimental values f_exp of the ground-state vibration frequencies. The comparison between the computational and experimental results gives a relative error about 1%, which is caused by the inexistence of an exact solution to the Schrödinger Equation (A1). The radial wave function R_n,J given by Equation (A13) is only a second-order approximation as described by Equation (A6). The approximation gets better as r is closer to the equilibrium bond length r₀ of the Morse potential.

Table 1. Parameters of four diatomic molecules and the comparison of the computed ground-state vibration frequencies f_QHM with the measured frequency f_exp.

Diatomic Molecules	H-H	H-Cl	O-O	N-N
Bond length r0 (m)	74 × 10−12	127.5 × 10−12	148 × 10−12	145 × 10−12
Reduced mass (kg/atom)	0.837 × 10−27	1.628 × 10−27	13.28 × 10−27	11.63 × 10−27
Potential width β (m−1)	1.94 × 1010	1.81 × 1010	2.67 × 1010	2.70 × 1010
ED (kg · m2· sec−2)	7.11 × 10−19	7.39 × 10−19	8.28 × 10−19	15.77 × 10−19
	16.8403	25.6722	52.7203	67.2966
T = 4π / (2λ − 1)	0.3845	0.2496	0.1203	0.09406
T = TM / (ħβ2)	8.091 × 10−15	11.74 × 10−15	21.31 × 10−15	14.24 × 10−15
fQHM = 1 / T	12,360 × 1010	8522 × 1010	4693 × 1010	7021 × 1010
Experiment fexp	12,470 × 1010	8652 × 1010	4666 × 1010	6987 × 1010
|fexp − fQHM| / fexp	0.8821%	1.503%	0.5786%	0.4866%

Table 1. Parameters of four diatomic molecules and the comparison of the computed ground-state vibration frequencies f_QHM with the measured frequency f_exp.

Diatomic Molecules	H-H	H-Cl	O-O	N-N
Bond length r0 (m)	74 × 10−12	127.5 × 10−12	148 × 10−12	145 × 10−12
Reduced mass (kg/atom)	0.837 × 10−27	1.628 × 10−27	13.28 × 10−27	11.63 × 10−27
Potential width β (m−1)	1.94 × 1010	1.81 × 1010	2.67 × 1010	2.70 × 1010
ED (kg · m2· sec−2)	7.11 × 10−19	7.39 × 10−19	8.28 × 10−19	15.77 × 10−19
	16.8403	25.6722	52.7203	67.2966
T = 4π / (2λ − 1)	0.3845	0.2496	0.1203	0.09406
T = TM / (ħβ2)	8.091 × 10−15	11.74 × 10−15	21.31 × 10−15	14.24 × 10−15
fQHM = 1 / T	12,360 × 1010	8522 × 1010	4693 × 1010	7021 × 1010
Experiment fexp	12,470 × 1010	8652 × 1010	4666 × 1010	6987 × 1010
|fexp − fQHM| / fexp	0.8821%	1.503%	0.5786%	0.4866%

The vibration period T given by Equation (4.12) can also be derived by using a state-dependent force constant. A quantum force constant is defined as the second-order derivative of the total potential V_Total (r) evaluated at the equilibrium position r_eq:

(4.13)

The resulting force constant is state-dependent by noting that V_Total depends on the three quantum numbers (n,J,m_J) as given by Equation (3.12). A specific V_Total has been expressed explicitly in Equation (4.8) for the case of n = J =0, and the substitution of V_Total into Equation (4.13) yields the force constant as:

(4.14)

Then the classical relation between the vibration period T and the force constant K gives:

(4.15)

which reproduces the result of Equation (4.12) derived by state-dependent molecular dynamics.

The above discussions of ground vibrational state reveal a remarkable observation that by replacing the internuclear potential with the total potential V_Total, the conventional MM turns out to be state-dependent MM, which then yields consistent results with QM. This observation applies to every quantum state of the molecule. In every quantum state Ψ_n,J,mJ, there is a total potential V_Total, which completely determines the molecular quantum dynamics within this state. Figure 4 plots the total potentials for the first three vibrational states and their accompanying rotational states. It can be seen that for a vibrational state with quantum number n, there are n + 1 sub-shells, within which rotational states with different quantum number J reside.

Figure 4. The variation of the total potential V_Total with respect to the two quantum numbers n and J. The lowermost points r_eq of V_Total correspond to the equilibrium bond lengths, and the curvatures of V_Total at r_eq correspond to the force constants of the molecule. This figure schematically illustrates how the bond length and the force constant change due to the change of quantum states.

Figure 4. The variation of the total potential V_Total with respect to the two quantum numbers n and J. The lowermost points r_eq of V_Total correspond to the equilibrium bond lengths, and the curvatures of V_Total at r_eq correspond to the force constants of the molecule. This figure schematically illustrates how the bond length and the force constant change due to the change of quantum states.

In excited vibrational states n ≥ 1, the quantum dynamics (4.1a) has multiple equilibrium points, and the total potential possesses multiple-shell structure. To illustrate this property, we consider the first excited state as an example. The corresponding radial wavefunction is given by:

(4.16)

and the radial dynamics is obtained from Equation (4.1a) as:

(4.17)

There are two stable equilibrium points in this state, which can be solved from the condition dz / dt = 0 as:

(4.18)

On the other hand, the total potential in the first excited state is given by Equation (3.12):

(4.19)

By evaluating the constant α at n = 1 with λ = 2 and b = 3, we have:

(4.20)

Substituting this α into Equation (4.19), the variation of V_Total (r,θ_eq) with respect to the angular quantum number J is plotted in Figure 4. It appears that there is a node at z = α + 1, where the total potential approaches to infinity. This infinite potential barrier separates V_Total (r,θ) into two shells. The lowermost point of the inner shell corresponds to the inner equilibrium point , while the lowermost point of the outer shell corresponds to the outer equilibrium point , as given by Equation (4.18). Also shown in Figure 4 is the total potential of the second excited state, which has three shells separated by the two nodes.

5. State-Dependent Orbital and Spin Dynamics

So far our discussions on molecular quantum dynamics are restricted to the radial vibration motion governed by Equation (4.1a). To take angular dynamics into account, all the three equations in Equation (4.1) have to be considered. In the previous sections we have seen that the QM descriptions of the quantum state Ψ_n,J,mJ can be reproduced in terms of the state-dependent molecular dynamics. Here, we go one-step further to show that the spin dynamics, which cannot be described by the spatial wave function Ψ_n,J,mJ, emerges naturally from the molecular dynamics (4.1) in the ground state where orbital angular momentum is zero.

Setting n = J = m_J = 0 and substituting R_0,0 (z) = e^−z/2z^α/2/r, Θ_0,0 (θ) = Φ₀ (ϕ) = 1 into Equation (4.1) yields the 3D quantum dynamics in the ground state:

(5.1)

where α and η are given by Equation (A14) with J = 0. We can see that even though the orbital angular quantum number J is set to zero, the angular velocity is actually not zero. The main reason why spin motion cannot be attributed to orbital motion in standard quantum mechanics is due to the definition of the orbital angular momentum L². To make this point more apparent, we inspect the expression of L² defined in Equation (4.2b) with the help of Equation (3.9b):

$$L^2={(M{r}^2\dot{\theta })}^2+\frac{{ħ}^2}{4}{\cot }^2\theta -{ħ}^2\frac{{\partial }^2\ln {\Psi }_{n,J,m_J}}{\partial {\theta }^2}+\frac{L_z^2}{{\sin }^2\theta }=J(J+1){ħ}^2$$

(5.2)

In case of J = m_J = 0, we have , ∂² lnΨ_n,J,mJ (r,θ,ϕ) / ∂θ² = ∂² lnR_n,J (r) / ∂θ² = 0 and the above equation is reduced to:

$${(M{r}^2\dot{\theta })}^2+({ħ}^2/4){\cot }^2\theta =0$$

(5.3)

In the real domain, the only solution is and θ = π / 2, which is the well-known spinless motion. However, in the complex domain, Equation (5.3) has a nontrivial solution:

$$\frac{d\theta }{dt}=-i\frac{ħ}{M{r}^2}\frac{\cot \theta }{2}$$

(5.4)

which is just the θ dynamics in Equation (5.1) in dimensionless form. In other words, the condition of L² = 0 does not completely nullify the complex-valued angular motion, and we can regard the spin dynamics as the remnant angular dynamics as the orbital angular momentum is zero.

The integration of the θ dynamics with θ = θ_R + iθ_I gives the three regions of trajectories as shown in Figure 5a. Within the Ω₀ region, the sign of dθ_R / dτ changes alternatively and produces zero mean velocity. Within the Ω₋ region, θ_R(τ) is monotonically decreasing with a mean value of dθ_R / dt equal to −1. Within Ω₊ region, θ_R (τ) is monotonically increasing with a mean value of dθ_R / dτ equal to one.

Figure 5. The complex θ(τ) trajectories for the states (a) J = 0, and (b) J = 1 with m_J = 0 solved from Equation (4.1) with r at its equilibrium position. The upper region Ω₋ contains spin-down dynamics with dθ_R / dτ < 0, the central region Ω₀ contains spinless dynamics exhibiting periodic motion around the equilibrium points and the lower region Ω₊ contains spin-up dynamics with dθ_R / dτ > 0.

Figure 5. The complex θ(τ) trajectories for the states (a) J = 0, and (b) J = 1 with m_J = 0 solved from Equation (4.1) with r at its equilibrium position. The upper region Ω₋ contains spin-down dynamics with dθ_R / dτ < 0, the central region Ω₀ contains spinless dynamics exhibiting periodic motion around the equilibrium points and the lower region Ω₊ contains spin-up dynamics with dθ_R / dτ > 0.

Although orbital angular momentum is completely depressed by the condition J = m_J = 0, non-zero angular motion does exist in the regions of Ω₋ and Ω₊. To relate this remnant angular motion to spin, the next step is to identify the magnitude of this remnant angular momentum. In the regions of Ω₋ and Ω₊, we note the following relations for the complex variable θ = θ_R + iθ_I:

cosθ = cosθ_Rcoshθ_I − isinθ_Rsinhθ_I = e^|θ_I|(cosθ_R ∓ isinθ_R) / 2, |θ_I| ≫ 0 ,

(5.5a)

sinθ = sinθ_Rcoshθ_I + icosθ_Rsinhθ_I = e^|θ_I|(sinθ_R ± icosθ_R) / 2, |θ_I| ≫ 0 .

(5.5b)

Therefore, angular momentum in the θ direction becomes:

(5.6)

In the region Ω₀, the θ(τ) dynamics exhibits periodic motion around the equilibrium points and yields zero mean angular velocity. According to the behavior of the θ dynamics, three spin regions can be defined,

(5.7)

The spin dynamics we discuss so far originates from the wavefunction Ψ = R_n,J (ρ)Θ_J,mJ (θ)Φ_mJ (ϕ) with Θ_J,mJ (θ) given by , the first-type Legendre function. Indeed, is only a special solution for Θ_J,mJ (θ) whose general solution can be represented by:

$${\Theta }_{J,m_J}(\theta )=B_1{P}_J^{m_J}(\cos \theta )+B_2{Q}_J^{m_J}(\cos \theta )$$

(5.8)

where is the second-type Legendre function. It can be shown by integration Equation (4.1) with that the direction of the angular motion produced by is always anti-parallel to that produced by . Therefore, in the same quantum state specified by the spatial quantum numbers (n,J,m_J), the molecule can behave either according to the spin dynamics with or to the anti-spin dynamics with . For the former case, we say that the molecule is in a sub-state of (n,J,m_J) denoted by (n,J,m_J,1/2), and for the latter case, the molecule is in the sub-state (n,J,m_J,−1/2). Furthermore, an entangled spin dynamics can be simulated according to the state-dependent molecular dynamics by expressing Θ_J,mJ(θ) as a linear combination of and as described by Equation (5.8).

In the ground state, we have seen that the orbital angular motion vanishes and the remnant angular motion in the θ direction emerges as the spin dynamics. We proceed to demonstrate that in excited states, orbital and spin motions coexist and both contribute to angular momentum. To examine the interaction between orbital and spin dynamics, we next consider quantum states with J = 1 and m_J = 0. Since the z-component angular momentum L_z is zero in these quantum states, we are interested in where the orbital angular momentum emerges and how it interacts with the spin angular momentum. The θ and ϕ dynamics for these quantum states are governed by Equations (4.1b) and (4.1c) as:

(5.9)

The trajectories of θ (t) on the complex θ_R − θ_I plane are illustrated in Figure 5b, where it can be seen that as in the ground state, three regions representing three angular motions come about. In the region Ω₋ (θ_I ≫ 0), θ_R is monotonically decreasing, while in the region Ω₊ (θ_I ≪ 0), θ_R is monotonically increasing. In the central region Ω₀, θ_R is a periodic function of time. In conjunction with the relation (5.5), the mean angular momentum of Equation (5.9) becomes:

(5.10)

Comparing with Equation (5.7), we find that in the regions of Ω₋ and Ω₊, the angular momentum contains an additional component ħ contributed from the orbital motion J = 1 apart from the spin angular momentum ħ/2, indicating that the orbital angular motion, in the states with J = 1, and m_J = 0, is produced solely by the θ dynamics because of .

For a quantum state with quantum number J ≠ 0 and m_J ≠ 0, the θ and ϕ dynamics can be expressed by:

(5.11)

Making use of Equation (5.5) once again, we obtain a general expression for the angular momentum in the θ and ϕ directions as:

(5.12)

In summary, in the Ω₀ region, one has zero-mean θ dynamics and a quantized z-component angular momentum m_Jħ, which is a well-known result in QM; while in the Ω₋ and Ω₊ regions, one has an angular momentum ±(J + 1/2)ħ in the θ direction and a zero-mean orbital angular momentum in the ϕ direction, which is otherwise unknown to QM.

The angular momentum given by Equation (5.12), which is derived from the state-dependent molecular dynamics, provides us with a trajectory-based method to determine the rotational energy of a diatomic molecule and its rotational spectrum. The most used formula of rotational energy takes the following form:

(5.13)

which treats the molecule as a rigid rotor with moment of inertia I₀ = Mr₀² / 2. A typical rotational spectrum consists of a series of peaks corresponding to transitions between adjacent levels satisfying ∆J = ±1:

$${\nu }_{\text{rigid}}=\frac{\Delta E_R}{hc}=\frac{B}{hc}(J+1)(J+2)-\frac{B}{hc}J(J+1)=2\overline{B}(J+1),\overline{B}=\frac{{ħ}^2}{\text{4}hc{I}_0}$$

(5.14)

The rigid-rotor model assumes a constant internuclear distance r₀ and neglects the stretch of the bond as the molecule rotates. To account for bond stretching due to rotation, we consider a refined rotation-energy component E_R in the eigenvalue expansion given by Equation (A20), from which the wavenumber of the transition from J + 1 to J can be derived as:

$${\nu }_{\text{non-rigid}}=2\overline{B}(J+1)-4\overline{D}{(J+1)}^3,\overline{D}=\frac{1}{hc{E}_D}{\left({ħ}^2/4b{I}_0\right)}^2$$

(5.15)

where D is a correction factor known as the centrifugal distortion constant. Table 2 lists the measured far infrared absorption spectrum (ν_exp) [29] for the diatomic molecule H-Cl and the predicted spectrum by the rigid model (ν_rigid) and the non-rigid model (ν_nonrigid) with B = 10.58 cm⁻¹ and D = 5.6 × 10⁻⁴ cm⁻¹ determined from the molecular parameters listed in Table 1.

Table 2. Comparisons of the four predictions ν_rigid, ν_nonrigid, ν_QHM and ν_fitting with the measured rotational spectrum ν_exp.

Transitions	νrigid (cm-1)	νnon-rigid (cm-1)	νQHM (cm-1)	νfitting (cm-1)	νexp (cm-1)
J = 0 → 1	21.16	21.16	20.80	20.88	20.88	0.383%
J = 1 → 2	42.32	42.30	41.57	41.74	41.74	0.407%
J = 2 → 3	63.48	63.42	62.29	62.58	62.58	0.463%
J = 3 → 4	84.64	84.49	82.93	83.38	83.32	0.468%
J = 4 → 5	105.8	105.52	103.47	104.14	104.13	0.634%
J = 5 → 6	126.96	126.48	123.87	124.83	124.73	0.689%
J = 6 → 7	148.12	147.35	144.11	145.45	145.37	0.867%
J = 7 → 8	169.28	168.13	164.16	165.97	165.89	1.024%
J = 8 → 9	186.65	188.81	184.01	186.40	186.23	1.192%

Table 2. Comparisons of the four predictions ν_rigid, ν_nonrigid, ν_QHM and ν_fitting with the measured rotational spectrum ν_exp.

Transitions	νrigid (cm-1)	νnon-rigid (cm-1)	νQHM (cm-1)	νfitting (cm-1)	νexp (cm-1)
J = 0 → 1	21.16	21.16	20.80	20.88	20.88	0.383%
J = 1 → 2	42.32	42.30	41.57	41.74	41.74	0.407%
J = 2 → 3	63.48	63.42	62.29	62.58	62.58	0.463%
J = 3 → 4	84.64	84.49	82.93	83.38	83.32	0.468%
J = 4 → 5	105.8	105.52	103.47	104.14	104.13	0.634%
J = 5 → 6	126.96	126.48	123.87	124.83	124.73	0.689%
J = 6 → 7	148.12	147.35	144.11	145.45	145.37	0.867%
J = 7 → 8	169.28	168.13	164.16	165.97	165.89	1.024%
J = 8 → 9	186.65	188.81	184.01	186.40	186.23	1.192%

Instead of using the quantum rotational energy obtained from the eigenvalues of Schrödinger equation, we will give an estimate of the wavenumber of transition based on the classical expression of the rotational energy with r and computed by the state-dependent molecular dynamics:

(5.16)

where is given by Equation (5.12) and the bond length r is denoted by r_J to emphasize its dependence on the angular quantum number J as given by Equation (4.5) and Equation (A14). An alternative estimate of the wavenumber caused by rotation absorption transition turns out to be:

(5.17)

With the molecular parameters of HCl given by Table 1, the bond length r_J is calculated by using Equation (4.5) and then substituted into Equation (5.17) to yield the estimated spectrum ν_QHM.

Table 2 compares the four predictions ν_rigid, ν_nonrigid, ν_QHM and ν_fitting with the measured rotational spectrum ν_exp. It can be seen that the wavenumber ν_fitting is closest to ν_exp, because ν_fitting is obtained by tuning the parameters B and D, so that Equation (5.15) best fits the experimental data. The curve fitting gives the best values as B* = 10.44 cm⁻¹ and D* = 5.2 × 10⁻⁴ cm⁻¹. The two wavenumbers ν_nonrigid and ν_QHM can be compared on the same footing, since both adopt the same molecular parameters listed in Table 1 in the computation. Their difference lies in the model of rotational energy used to compute wavenumbers. The wavenumber ν_nonrigid is based on the eigen-energy model (A20), while ν_QHM is based on the QHM model (5.16). Table 2 reveals that the wavenumber ν_QHM is much closer to the experimental results than ν_nonrigid.

Theoretically, the description of state-dependent molecular dynamics by QHM is equivalent to QM; however, the actual accuracy depends on the degree of approximation involved in the solution to the wave function. The equilibrium bond length r_J computed from Equation (4.5) is based on the radial wave function R_n,J given by Equation (A13), which is an approximate solution to the radial Schrödinger Equation (A5) by employing the second-order expansion (A6). When the quantum number J increases, the bond length deviates further from the equilibrium position r₀ of the Morse potential, and the accuracy of the expansion (A6) gets worse. This tendency explains the degradation of the prediction accuracy of ν_QHM with respect to the measured rotational spectrum ν_exp.