Quantum–Classical Mechanics and the Franck–Condon Principle ^†

Abstract

Quantum–classical mechanics and the Franck–Condon principle related to quantum mechanics are discussed as two alternative theoretical approaches to molecular optical spectroscopy. The statement of the problem is connected with the singularity of quantum mechanics in describing the joint motion of electrons and nuclei in the transient state of “quantum” transitions. This singularity can be eliminated by introducing chaos into the transient state. Quantum mechanics itself, supplemented by chaos (dozy chaos), is called quantum–classical mechanics. Using the simplest example of quantum transitions, it is shown that the results of quantum–classical mechanics in the case of strong dozy chaos correspond to the physical picture based on quantum mechanics and the Franck–Condon principle. The same chaos can be strong for small molecules in standard molecular spectroscopy and simultaneously weak in the photochemistry and nanophotonics of large molecules, where quantum mechanics no longer works. To describe the chaotic dynamics of the transient state, it is necessary to apply quantum–classical mechanics. Thus, the erroneous Franck–Condon physical picture of molecular “quantum” transitions is workable from a practical point of view as long as we are dealing with sufficiently small molecules, just as the erroneous geocentric picture of the world was workable until we went out into outer space.

1. Introduction

As is known, quantum mechanics is inextricably linked with classical mechanics. Its justification is connected with the need to consider the interaction of a microparticle with a macroscopic classical measuring device [1]. The basic dynamical equation, the Schrödinger equation, was postulated by Schrödinger but actually derived from the Hamilton–Jacobi equation for the action S in classical mechanics by introducing the wave function in some form, which is now referred to as semiclassical approximation. The width of the levels “inside which” the energy spectrum is continuous is a sign of the partially classical nature of the dynamics in quantum systems. Quantum–classical mechanics is not a “mixture” of quantum mechanics and classical mechanics, but it does involve substantially modified quantum mechanics in which the initial and final states are quantum in the Born–Oppenheimer adiabatic approximation [2,3,4,5,6] and the chaotic transient state, due to chaos, is classical. In molecular physics, the Franck–Condon principle [7,8,9,10] avoids the consideration of transient state dynamics, which is unreasonably assumed to be unimportant. Classicality, which is immanently inherent in quantum mechanics itself, in molecular physics, is supplemented by classicism, which is associated with the Franck–Condon principle. It is assumed that the quantum transition (fast vertical jump) of an electron from the ground to the excited electronic state of the molecule occurs between the turning points of classically moving nuclei, where the nuclei are at rest. In fact, the classical nature of motion in molecular physics is not associated with the Franck–Condon principle but with the chaotic dynamics of the motion of an electron and nuclei in a transient state.

The theory of quantum transitions in quantum mechanics [11] successfully describes quantum transitions in atomic and nuclear physics. In molecular physics, a “successful description” is provided only under the conditions of applying the Born–Oppenheimer adiabatic approximation and the Franck–Condon principle. If the Born–Oppenheimer adiabatic approximation is not met, quantum mechanics becomes singular. Specifically, the series of time-dependent perturbation theory diverges. Obviously, in real molecules, the adiabatic approximation is not strictly observed, which makes the application of the Franck–Condon principle unfounded in theory, along with the whole physical picture of the molecular transitions based on it. The only physical way to eliminate the singularity of the series of time-dependent perturbation theory in molecular physics is to postulate the presence of dynamics in the transient electron-nuclear(-vibrational) state, which the Franck–Condon principle ignores, and this dynamic is chaotic. In the case of strong chaos, as in the case of the Franck–Condon picture of molecular transitions, the transition rates do not depend on the specific dynamics of the transient state but instead depend only on the initial and final states taken in the adiabatic approximation. In the case of weak chaos, against the background of chaos, the regular nature of the dynamics of the transient state manifests itself. Chaos, which is weak in the case of large molecules, may be strong in the case of small molecules. Therefore, the Franck–Condon picture of transitions often gives good agreement with experimental data on optical spectra in conventional molecular spectroscopy [12,13,14] studies of small molecules. In photochemistry and nanophotonics, where, as a rule, we deal with large molecules, where chaos is not strong but weak, elements of dynamic self-organization often appear in the chaotic dynamics of the transient state. A striking example of this is the well-known narrow and intense J-band of J-aggregates of polymethine dyes [15,16,17,18,19,20,21,22,23,24], which can no longer be explained on the basis of quantum mechanics but can be explained in quantum–classical mechanics as Egorov nano-resonance [25,26,27,28,29,30,31,32,33,34]. Thus, in the case of small molecules, the Franck–Condon principle gives the correct result, although an erroneous theory and an erroneous physical picture are used. In the case of large molecules, this erroneous theory and the erroneous physical picture no longer lead to the correct result. The analog of this situation is well known.

This analog is the collision between two outlooks of the world: geocentric and heliocentric. As is known, the correct outlook is the heliocentric outlook of the world, in which the Earth rotates both around the Sun and around its own axis. However, being on the surface of the Earth, the rotation of the Earth around its own axis is perceived by the observer as the movement of the Sun across the sky, which is adequately simulated by the erroneous geocentric outlook. It is even customary to talk about the time of sunrise and the time of its sunset at a given particular point on the surface of the Earth. However, the exit from the surface of the Earth to a sufficiently large distance into space directly shows the fallacy of the geocentric view of the world.

At present, quantum–classical mechanics and the physical picture of molecular “quantum” transitions corresponding to it are based on their simplest example, namely, the example of quantum–classical mechanics of elementary electron transfers in condensed media [31,35]. The appearance of chaos in the transient state (dozy chaos [28,29]) is achieved by a significant modification of the total Green’s function of the system as a result of replacing the known infinitesimal $\gamma$ in its energy denominator with a finite $\gamma$ [27,31,35].

It is shown below that the result of quantum–classical mechanics of elementary electron transfers for the case of strong dozy chaos corresponds to the result based on quantum mechanics and the Franck–Condon principle. Also, using the example of the transformation of the shape of the optical absorption band in quantum–classical mechanics, it is shown how, with a decrease in the extent of the quantum–classical transition, the result of quantum–classical mechanics, where dozy chaos is rather weak, passes into the result of quantum mechanics, where the same dozy chaos is already strong enough.

2. Quantum–Classical Mechanics of Elementary Electron Transfers in Condensed Media

In a sense, the quantum–classical mechanics of elementary electron transfers in condensed media [31,35] is a generalization of the theory of multi-phonon transitions [3] based on quantum mechanics, the Born–Oppenheimer adiabatic approximation, and the Franck–Condon principle, where the simplest problem solved was the problem regarding the shape of optical bands, which arose as a result of multi-phonon transitions in F-centers [36,37,38,39]. It was in the case of this problem of electron transfers to a distance $L\equiv \left|L\right|$ of the order of a nanometer that the Egorov nano-resonance (see Section 1 and Section 3) was discovered [30,31,32]. Compared to the simplest Hamiltonian, which was used in the F-center problem, our Hamiltonian is complicated only by an additional electron potential well [3,28,29,30,31,32,35]:

$$H=-\frac{{\hslash }^2}{2m}{\Delta }_r+V_1\left(r\right)+V_2\left(r-L\right)+{\displaystyle \sum _{\kappa }{V}_{\kappa }\left(r\right) q_{\kappa }}+\frac{1}{2}{\displaystyle \sum _{\kappa }\hslash {\omega }_{\kappa }\left(q_{\kappa }^2-\frac{{\partial }^2}{\partial q_{\kappa }^2}\right)}.$$

(1)

Comments on this Hamiltonian can be found in [27]. Similarly, in [27], one can find comments on the nuclear reorganization energy

$$E=\frac{1}{2}{\displaystyle \sum _{\kappa }\hslash {\omega }_{\kappa }{\tilde{q}}_{\kappa }^2}.$$

(2)

An attempt to take into account the transient state dynamics using the Green’s function method in quantum mechanics with Hamiltonian (1) leads to significant singularity in the rates of “quantum” transitions [30,31,32].

This singularity is eliminated by replacing the infinitesimally small $\gamma$ with a finite quantity [28,29,30,31,32,33,34,35,40] in the Green’s function

$$G_H\left(r,r^{\prime }; q,q^{\prime }; E_H\right)={\displaystyle \sum _s \frac{{\mathsf{\Psi }}_s\left(r,q\right) {\mathsf{\Psi }}_s\ast \left(r^{\prime },q^{\prime }\right)}{E_H-E_s-i\gamma }},$$

(3)

where ${\mathsf{\Psi }}_s\left(r,q\right)$ are the eigenfunctions of the Hamiltonian (1); $\left(r,q\right)$ are all electronic and nuclear coordinates; $E_s$ are the eigenvalues of $H$; and $E_H$ is the total energy of the system. The aforementioned replacement of $\gamma$ with a finite quantity in Equation (3), or in similar Green’s functions of more complex molecular and/or condensed matter systems, lies at the foundation of a new theory—dozy-chaos or quantum–classical mechanics [35] (cf. Section 1).

Comparison of theory with experiment shows that $\gamma >>\hslash \omega$ (${\omega }_{\kappa }=\mathrm{constant}\equiv \omega$; see below, Section 3) [25,26,28,29,30,31,32,33,34,40]. The dozy-chaos energy $\gamma$ is a measure of dynamic or dozy chaos in the transient state [28,29] (see Section 4). The physics of dozy chaos is discussed in [35,41,42,43]. The physics can be clarified only after creating quantum–classical electrodynamics, which will be created in future similarly to how quantum electrodynamics [44,45,46] was created in due time after the creation of quantum mechanics [1,11].

There is an obvious analogy in the history of the discovery of the elementary quantum of action, Planck’s constant $\hslash$ [47], and the discovery of the dozy-chaos energy $\gamma$ [28,29,30,31,32,35,41,43]. However, $\hslash$ is a universal physical constant, while $\gamma$ is not a universal physical constant, and its value depends on a specific physical system. As a comparison of theory with experiment shows, the numerical value of the energy $\gamma$ is not completely indefinite but approximately ranges from $\gamma \cong E$ (strong dozy chaos) to $\hslash \omega <<\gamma <<E$ (weak dozy chaos).

If we keep in mind the inseparable connection between quantum mechanics and classical mechanics, which was noted above (see Section 1), then we can say that Planck’s constant $\hslash$ is a hidden parameter in classical mechanics, and dozy-chaos energy $\gamma$ is a hidden parameter in quantum mechanics.

3. Shape of the Optical Absorption Band and Egorov Nano-Resonance (Enr)

The quantum–classical mechanics of elementary electron transfers in the framework of the Einstein model ${\omega }_{\kappa }=\mathrm{constant}\equiv \omega$ gives the following analytical result for absorption band shapes $K=K\left(\mathsf{\Omega }\right)$ (Equations (4)–(24), $\mathsf{\Omega }$ is the frequency of light) [28,29,30,31,32,33,34,35,40]:

$$K=K_0\exp W,$$

(4)

$$\begin{gathered} W=\frac{1}{2}\ln \left(\frac{\omega \tau \sin \mathrm{h} {\beta }_T}{4\pi \cos \mathrm{h} t}\right)-\frac{2}{\omega \tau }\left(\cot \mathrm{h} {\beta }_T-\frac{\cos \mathrm{h} t}{\sin \mathrm{h} {\beta }_T}\right) \\ +\left({\beta }_T-t\right)\frac{1}{\omega \tau \mathsf{\Theta }}-\frac{\sin \mathrm{h} {\beta }_T}{4\omega \tau {\mathsf{\Theta }}^2\cos \mathrm{h} t}, \end{gathered}$$

(5)

$$1<<\frac{1}{\omega \tau \mathsf{\Theta }}\le \frac{2\cos \mathrm{h} t}{\omega \tau \sin \mathrm{h} {\beta }_T},$$

(6)

where ${\beta }_T\equiv \hslash \omega /2k_{\mathrm{B}}T$, $T$ is the absolute temperature,

$$t=\frac{\omega {\tau }_{\mathrm{e}}}{\theta }\left[\frac{AC+BD}{A^2+B^2}+\frac{2\mathsf{\Theta }\left(\mathsf{\Theta }-1\right)}{{\left(\mathsf{\Theta }-1\right)}^2+{\left(\mathsf{\Theta }/{\theta }_0\right)}^2}+\frac{{\theta }_0{}^2}{{\theta }_0{}^2+1}\right],$$

(7)

$$\left|{\theta }_0\right|>>\frac{E}{2J_1},$$

(8)

$$\theta \equiv \frac{{\tau }_{\mathrm{e}}}{\tau }=\frac{L E}{\hslash \sqrt{2J_1/m}}, \mathsf{\Theta }\equiv \frac{{\tau }^{\prime }}{\tau }=\frac{E}{\Delta }, {\theta }_0\equiv \frac{{\tau }_0}{\tau }=\frac{E}{\gamma },$$

(9)

$${\tau }_{\mathrm{e}}=\frac{L}{\sqrt{2J_1/m}}, \tau =\frac{\hslash }{E}, {\tau }^{\prime }=\frac{\hslash }{\Delta }, {\tau }_0=\frac{\hslash }{\gamma }.$$

(10)

Here, we use the notation

$$A=\cos \left(\frac{\theta }{{\theta }_0}\right)+\Lambda +{\left(\frac{1}{{\theta }_0}\right)}^2{\rm N},$$

(11)

$$B=\sin \left(\frac{\theta }{{\theta }_0}\right)+\frac{1}{{\theta }_0}{\rm M},$$

(12)

$$C=\theta \left[\cos \left(\frac{\theta }{{\theta }_0}\right)-\frac{1-{\xi }^2}{2{\theta }_0}\sin \left(\frac{\theta }{{\theta }_0}\right)\right]+{\rm M},$$

(13)

$$D=\theta \left[\sin \left(\frac{\theta }{{\theta }_0}\right)+\frac{1-{\xi }^2}{2{\theta }_0}\cos \left(\frac{\theta }{{\theta }_0}\right)\right]-\frac{2}{{\theta }_0}{\rm N},$$

(14)

and

$$\xi \equiv {\left(1-\frac{E}{J_1}\right)}^{1/2} (J_1>E \mathrm{by} \mathrm{definition}),$$

(15)

and where we finally have

$$\Lambda =-{\left(\mathsf{\Theta }-1\right)}^2{\rm E}+\left[\frac{\left(\mathsf{\Theta }-1\right)\theta }{\rho }+\mathsf{\Theta }\left(\mathsf{\Theta }-2\right)\right]{{\rm E}}^{\frac{1-\rho }{1-\xi }},$$

(16)

$${\rm M}=2\mathsf{\Theta }\left(\mathsf{\Theta }-1\right){\rm E}-\left[\frac{\left(2\mathsf{\Theta }-1\right)\theta }{\rho }+2\mathsf{\Theta }\left(\mathsf{\Theta }-1\right)\right]{{\rm E}}^{\frac{1-\rho }{1-\xi }},$$

(17)

$${\rm N}=\mathsf{\Theta }\left[\mathsf{\Theta }{\rm E}-\left(\frac{\theta }{\rho }+\mathsf{\Theta }\right){{\rm E}}^{\frac{1-\rho }{1-\xi }}\right],$$

(18)

$${\rm E}\equiv \exp \left(\frac{2\theta }{1+\xi }\right), \rho \equiv \sqrt{{\xi }^{ 2}+\frac{1-{\xi }^{ 2}}{\mathsf{\Theta }}}.$$

(19)

The factor $K_0$ becomes

$$K_0=K_0^e{K}_0^p,$$

(20)

where

$$K_0^e=\frac{2{\tau }^3{J}_1}{m}\frac{\left(A^2+B^2\right){\rho }^3{\mathsf{\Theta }}^4\xi }{{\theta }^2{\left[{\left(\mathsf{\Theta }-1\right)}^2+{\left(\frac{\mathsf{\Theta }}{{\theta }_0}\right)}^2\right]}^2\left[1+{\left(\frac{1}{{\theta }_0}\right)}^2\right]}\cdot \eta ,$$

(21)

and

$$\eta \equiv \exp \left(-\frac{4\theta }{1-{\xi }^2}\right),$$

(22)

and

$$K_0^p=\frac{1}{\omega \tau }{\left[1+\frac{\sin \mathrm{h}\left({\beta }_T-2t\right)}{\sin \mathrm{h} {\beta }_T}\right]}^2+\frac{\cos \mathrm{h}\left({\beta }_T-2t\right)}{\sin \mathrm{h} {\beta }_T}.$$

(23)

In Equations (9) and (10), $\Delta >0$ is the heat energy associated with the energy $\hslash \mathsf{\Omega }$ of the absorbed photon by the law of energy conservation:

$$\hslash \mathsf{\Omega }=J_1-J_2+\Delta ,$$

(24)

where $J_1$ and $J_2$ are the electron binding energy at donor 1 and acceptor 2.

The time scales given by Equation (10) control the chaotic dynamics of the transient state of elementary electron transfers in condensed media. They are discussed in [28,29,32,33,35,48,49,50]. The donor–acceptor distance L is also equal to the length of the polymethine chain, the main optical chromophore of polymethine dyes, where the electronic charge on carbon atoms alternates along the chain and alternatively redistributes upon optical excitation; therefore, in Equations (21) and (22), we can often take $\eta \approx 1$ [28,29,30,31,32,33,34,40].

Given by Equation (10), the time scales

$${\tau }_{\mathrm{e}}=\frac{L}{\sqrt{2J_1/m}}$$

(25)

and

$$\tau =\frac{\hslash }{E}$$

(26)

are included in the Egorov nano-resonance (see Section 1) [25,26,27,28,29,30,31,32,33,34] as follows:

$${\left(2{\tau }_e\right)}^{-1}={\tau }^{-1}.$$

(27)

4. From Quantum–Classical Mechanics to Quantum Mechanics

Just as there is no strict limit transition from quantum mechanics to classical mechanics, there is no strict limit transition from quantum–classical mechanics to quantum mechanics. The standard result in the theory of many-phonon transitions [3] is effected from Equations (4)–(24) by $\gamma \to \infty$ (${\theta }_0\to 0$ according to Equation (9)) in Equation (7) for $t$ (see Figure 3 in [35]) and by $\gamma \to 0$ (${\theta }_0\to \infty$) in Equation (21) for $K_0^e$ (see Equation (162) in [35]) [31,35]:

$$K=\frac{a^2\hslash }{\sqrt{4\pi {\lambda }_1{k}_{\mathrm{B}}T}}\exp (-\frac{2L}{a})\exp \left[-\frac{{\left(\Delta -{\lambda }_{\mathrm{r}}\right)}^2}{4{\lambda }_{\mathrm{r}}{k}_{\mathrm{B}}T}\right]$$

(28)

where $a\equiv \hslash /\sqrt{2m{J}_1}$ and ${\lambda }_{\mathrm{r}}\equiv 2E$. The Gaussian formula (Marcus formula)

$$K=K(\Delta )\propto \exp \left[-\frac{{\left(\Delta -{\lambda }_{\mathrm{r}}\right)}^2}{4{\lambda }_{\mathrm{r}}{k}_{\mathrm{B}}T}\right]$$

is the simplest result in quantum–mechanical theory at high (room) temperatures (see [3,4]). This result was obtained by Marcus [51,52,53,54,55,56] and even earlier by Huang and Rhys [57], Pekar [36,37,38], Lax [58], Krivoglaz and Pekar [39], and Krivoglaz [59]. In contrast to Equations (4)–(24), Equation (28) does not take into account the chaotic dynamics of the transient state.

The main features of the result (4)–(24) of quantum–classical mechanics are easy to understand from a potential box with a movable wall which simulates nuclear reorganization (Figure 1).

In Figure 1b, the shift of the optical band peak to the red region and its narrowing with decreasing $\gamma$ is explained by the decrease in friction at the base of the moving wall (see Figure 1a); higher wall mobility leads to a greater decrease in the effective gap between the ground and excited electronic energy levels and to a higher the degree of organization in the quantum–classical transition.

The fact that there is no rigorous transition from quantum–classical mechanics to quantum mechanics indicates the fundamental novelty of quantum–classical mechanics, which is fundamentally not reduced to quantum mechanics [27].

5. Extent of Quantum–Classical Transitions as a Measure of Dozy Chaos

From general considerations, one can expect that the weak chaos of any physical nature in large systems can turn out to be strong chaos for small systems. Therefore, one would expect dozy chaos to be no exception in this respect. In other words, the weak dozy chaos in long-extended quantum–classical transitions can look like strong dozy chaos for short-extended quantum–classical transitions. Figure 2 demonstrates this. The Gaussian shape of the optical band (Figure 2, black graph), obtained as a result of the “limit transition” from quantum–classical mechanics to quantum mechanics, i.e., as a result of the limiting transition of dozy-chaos energy $\gamma$ to infinity in the exponent (see details in Section 4), corresponds to extremely strong dozy chaos. This means that if, for example, with a decrease in the sufficiently large length of the polymethine chain L (a decrease in the extent of the quantum–classical transition) and the constancy of all other parameters of the system, including a sufficiently small value of $\gamma$, corresponding to weak dozy chaos ($\gamma <<E$, see Section 2), as a result, we obtain a Gaussian-like band, then this band corresponds to effectively strong dozy chaos at small L. Figure 2 (colored graphs) demonstrates the transformation of the shape of the optical band as a result of a decrease in the extent of the quantum–classical transition, which leads to such a transient state with effectively strong dozy chaos (magic graph).

The shift of the Gaussian-like band (magic) to the red region and its narrowing with respect to the Gaussian band (black) from quantum mechanics is explained by the presence of a chaotic transient state in quantum–classical mechanics, which, by definition, is absent in quantum mechanics. The presence of a chaotic transient state leads to some effective reorganization energy of lesser energy E; a part of the reorganization energy E is represented in quantum–classical mechanics as a chaotic, dynamic transient state reorganization energy, which is formally expressed in dozy-chaos energy $\gamma$. Another way of interpreting the discussed effects originates from the consideration of a potential box with a movable wall (Figure 1a). We can say that the moving wall, which simulates nuclear reorganization in quantum–classical mechanics, has a much smaller “mass” compared to the “mass” of a fixed wall in quantum mechanics. If we correlate the “mass” of the wall with the value of the reorganization energy, similarly to how we correlated the friction value at the wall base with the value of dozy-chaos energy above, then, here, the discussed effects find their analogous explanation at the qualitative level [26,27]. Another qualitative explanation of the ‘‘redshift’’ and narrowing with decreasing nuclear reorganization energy is based on the use of standard theory and can be found in [40] (see Figures 3 and 4 therein).

Thus, using the example of the transformation of the shape of the optical absorption band, it is shown that, with a decrease in the extent of the quantum–classical transition, the result of quantum–classical mechanics, where the dozy chaos is sufficiently weak, essentially passes into the result of quantum mechanics, where the same dozy chaos is already sufficiently strong. Therefore, as a measure of dozy chaos, one can consider not only the dozy-chaos energy $\gamma$ but also the extent of quantum–classical transitions L.

6. Conclusions

It is shown that, in quantum–classical mechanics, in the case of strong dozy chaos, which is almost always realized for sufficiently small molecules, the rates of quantum–classical transitions do not depend on the specific dynamics of the transient state but instead depend only on the initial and final states. As is known, we have the same result in the standard Franck–Condon picture obtained in the framework of quantum mechanics (see, e.g., [11]). Thus, we can obtain an explanation of why the erroneous standard Franck–Condon picture [7,8,9,10] (see Section 1), in the case of sufficiently small molecules, is able to interpolate experimental data [12,13,14] well in standard molecular spectroscopy. We note that a sophisticated researcher, especially a theoretical physicist, should not be confused by the conclusion that an erroneous theory and an erroneous physical picture can often agree with experimental data. For example, the erroneous Aristotelian geocentric picture of the world has existed in human civilization for hundreds of years and is quite suitable from a practical point of view. If we ignore the genius of Galileo and other innovators like him, then the need for the appearance of a correct, namely heliocentric, picture of the world arose in practice only with the release of humankind into space. A similar need arises in our time in the quantum physics of molecular physico-chemical systems and in the quantum physics of condensed matter. Specifically, this need is expressed in the creation and development of quantum–classical mechanics in their exhaustive completeness to account for the chaotic dynamics of the transient state in extended systems.