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Nuclear Potential Energy Curves

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    Arriving at solutions of many complex quantum mechanical systems have posed great challenges to the scientists of both the past and the present. Few quantum systems have been solved analytically and even fewer without the help of approximations. The goal of this overview is to review the fundamentals of solving one of these complicated systems, the diatomic molecule.


    In contrast to atoms, diatomics show additional degrees of freedom of nuclear motion, e.g. vibrations and rotations, along with the motion of electrons. Inclusion of this additional degree of freedom complicates the system by making the system a three (or more depending on the number of electrons) body system, the solution of which has only been solved with approximation techniques (e.g. mean field theory or ignoring weak couplings). More specifically, we will define such properties as the nuclear potential curves and electronic energy curves for diatomic and then address the connection between these curves under the validity of the Born-Oppenheimer approximation and adiabatic approximation. In addition, breakdowns of Born-Oppenheimer approximation will be discussed.

    Neglecting higher order relativistic interactions, the total Hamiltonian for a diatomic system is

    \[\left[ -\sum_{\alpha = 1}^2\dfrac{\hbar^2}{2M_{\alpha}}\bigtriangledown_{\alpha}^{2} - \sum_{i - 1 }^n\dfrac{\hbar^2}{2m}\bigtriangledown_{i}^{2} + \dfrac{Z_{\alpha}Z_{\beta}e^2}{4\pi\epsilon_o \mid R_{\alpha} - R_{\beta}\mid} - \sum_{\alpha = 1}^2 \sum_{i = 1}^n \dfrac{Z_{\alpha}e^2}{4\pi\epsilon_o \mid R_{\alpha} - r_i \mid} + \sum_{i = 1}^{n} \sum_{j > 1}^n \dfrac{e^2}{4\pi\epsilon_o \mid r_i - r_j \mid} + \hat{H}_{SO} \right] \Psi (Q,q) = E\Psi (Q,q) \label{1}\]

    where the first and second summations represent the kinetic energy of the nuclei, \(\hat{T}_N\), and electrons, \(\hat{T}_E\), respectively. The next three terms express the potential energy of the system, \(V_{NE}\): the nuclei-nuclei repulsion, the electron-nuclei attraction and the electron-electron repulsion respectively. The last term is the spin-orbit operator. Note that the system wavefunction naturally includes both nuclear and electronic positions and that the Hamiltonian is not separable into nuclear and electronic motion, i.e., the motions of the electrons and nuclei are coupled by the fourth term. The total Hamiltonian can then be expressed as:

    \( \; \; \hat{H} = \hat{T}_{N} + \hat{H}_{el} + \hat{H}_{SO} \label{2}\)

    where the nuclei-nuclei repulsion term as well as the nuclei-electronic attractive and electronic-electronic repulsive terms is included in the electronic Hamiltonian:

    \[\displaystyle \; \hat{H}_{el} = -\sum_{i - 1 }^n\dfrac{\hbar^2}{2m}\bigtriangledown_{i}^{2} + \dfrac{Z_{\alpha}Z_{\beta}e^2}{4\pi\epsilon_o \mid R_{\alpha} - R_{\beta}\mid} - \sum_{\alpha = 1}^{2} \sum_{i = 1}^{n}\dfrac{Z_{\alpha}e^2}{4\pi\epsilon_o \mid R_{\alpha} - r_i \mid} + \sum_{i = 1}^{n} \sum_{j > 1}^{n}\dfrac{e^2}{4\pi\epsilon_o\mid r_i - r_j \mid} \label{3}\]

    Should the Hamiltonian, indeed, be made separable, then the corresponding wavefunction can be expressed as the product of the eigenfunctions of the nuclear and electronic motion. To do this, the nuclear/electronic coupling summation in Equation \(\ref{1}\) would have to be either ignored or the approach to solving the system be modified.

    Nuclear-Electronic Motion

    In order to solve this problem, we expand the total wavefunction for the system \(\Psi (Q,q)\) an assumed complete and orthonormal basis of adiabatic electronic wavefunctions. The intuitive basis of modifying this expansion lies in the relative velocity of the electrons versus the nuclei. Electron velocities are approximately ~3000 km/sec whereas the nuclei velocities are three orders of magnitude slower at around 3 km/sec. Such a disparaging difference in velocities implies that the electrons can be assumed to respond “instantaneously” to the motion of the nuclei and hence the electronic motion can be described as the motion of the electrons within the field of stationary nuclei, i.e. nuclear positions are parameters for solving the electronic wavefunction, not variables as are the electronic positions. The term adiabatic is used to describe systems as such. Hence the molecular wavefunction can be expressed as:

    \[ \; \; \Psi (Q,q) = \sum_{k}^{} a_k \mid x_k (Q) \rangle \mid \phi_{k} (q;Q)\rangle \label{4}\]

    where \(\mid x_k \rangle\) is the nuclear wavefunction that describes the motion of the nuclei on the potential energy surface associated with the kth adiabatic electronic state \(\mid \phi_k \rangle\). This expansion shows that nuclear motion occurs on all electronic states simultaneously. A common approximation is to assume that this expansion can be represented by one term in the summation, with only one ak coefficient being non-zero, and hence nuclear motion occurs on only one electronic state. This approximation is the Born-Oppenheimer approximation and when a wavefunction can be represented as such a product it is called a Born-Oppenheimer state. The adiabatic electronic eigenstates used in the previous expansion are calculated from solving the Schrödinger equation with the electronic Hamiltonian from Equation \(\ref{3}\):

    \[ \; \; \hat{H}_{el} \mid \phi_k(q;Q)\rangle = \epsilon_k(Q) \mid \phi_k (q;Q)\rangle \label{5}\]

    Note that the adiabatic electronic eigenstates, \(\mid \phi_k \rangle\) depend parametrically on the values of the nuclear positions. Additionally, \(\epsilon_k(Q)\), which represent the adiabatic electronic energy curve of the kth electronic eigenstate, varies with the values of the nuclear positions. By substituting this expanded wavefunction into the Schrödinger equation with total Hamiltonian (Equation \(\ref{3}\)), then multiplying by the kth adiabatic wavefunction and integrating over electronic coordinates we obtain an infinite set of coupled equations:

    \[\displaystyle \; \; a_k \left[\hat{T}_N + T^{'}_{kk} + T^{''}_{kk} + \epsilon_k(Q) + SO_{kk} - E \right] |\chi_k(Q)\rangle = -\sum_{k^{'}\not= k}^{} a_{k'} \left[T^{'}_{kk'} + T^{''}_{kk'} + SO_{kk'}\right] | \chi_k (Q)\rangle \label{6}\]


    \[ T^{''}_{kk'} = \dfrac{-\hbar^2}{2\mu_R} \langle \phi_k \mid \bigtriangledown^{2}_{R} \mid \phi_{k'} \rangle \label{7}\]

    \[T^{''}_{kk'} = \dfrac{-\hbar^2}{\mu_R} \langle \phi_k \mid \bigtriangledown_R \mid \phi_{k'} \rangle \cdot \bigtriangledown_R\label{8}\]

    \[SO_{kk'} = \langle \phi_k \mid \hat{H}_{SO} \mid \phi_{k'} \rangle \label{9}\]

    where the nuclear kinetic energy operator can be expanded in a center of mass term and a internal motion term, where \(R\) is the internuclear distance and \(\mu\) is the reduced mass:

    \[-\sum^{2}_{\alpha = 1} \dfrac{\hbar^2}{2M_{\alpha}} \bigtriangledown^2_{\alpha} = - \dfrac{\hbar^2}{2M} \bigtriangledown^2_{cm} - \dfrac{\hbar^2}{2\mu_{\alpha}} \bigtriangledown^2_R \label{10}\]

    The center of mass has been omitted in Equation \(\ref{6}\) for clarity.

    The Born-Oppenheimer Approximation and Adiabatic Approximation

    The most common and easiest approach to solving Equation \(\ref{6}\) is to assume that the wavefunction can be represented with one term in the expansion of Equation \(\ref{4}\). This approximation thus assumes that the motion of the electrons do not depend on the motion of the nuclei, just the nuclear positions. Another interpretation of this approximation is that nuclear motion occurs on only one electronic energy curve and hence in only a single electronic state. Max Born and his student Robert Oppenheimer demonstrated that under many situations the coupling terms on the right side of Equation \(\ref{6}\) can be ignored, thus simplifying the problem immensely by converting a large coupled system into two simpler uncoupled systems. Hence this approximation is commonly called the Born-Oppenheimer (BO) approximation. Often this approximation is referred to as the adiabatic approximation. The BO approximation will then be a further simplification of the problem (vida infra). Thus the total wavefunction can be expressed as only one term in Equation \(\ref{4}\), then the right side of Equation \(\ref{6}\) is zero since all other ak’s are zero. Thus we obtain the Schrödinger’s equation for the motion of the nuclei under the adiabtic approximation:

    \[ \left[\hat{T}_N + V_R(Q) - E_{TOT}\right]\mid \chi_K(Q) \rangle = 0 \label{11}\]

    where is the effective nuclear potential energy operator that governs the motion of the nuclei and is given by:

    \[V_R(Q) = T^{'}_{kk} + T^{''}_{kk} + \epsilon_k(Q) + SO_{kk} \label{12}\]

    Note that nuclear potential energy is a function of both the nuclear position and nuclear momentum. The Born Oppenheimer approximation further simplifies the problem by neglecting the \(T^{'}_{kk}\) and \( T^{''}_{kk}\) terms above with the following rational. The term is usually small as the electronic eigenstates do not normally oscillate greatly with internuclear separation. Equally, when nuclear motion is slow and the adiabatic wavefunction doesn’t oscillate greatly, the \(T^{'}_{kk}\) can be neglected. So in the absence of appreciable spin-orbit coupling, the effective nuclear potential energy curve is the adiabatic electronic energy curve. These additional terms in Equation \(\ref{12}\) represent non-adiabatic corrections to the nuclear potential energy curves but the adiabatic approximation of nuclear motion still occurring in only one electronic energy state holds. In this BO situation, where the nuclear and electronic motions are separable, the system can be solved in two stages. The first stage involves solving Equation \(\ref{5}\) for the adiabatic electronic state \(\mid \phi_k \rangle\) and energy curve \(\epsilon_{k}(Q)\)that nuclear motion occurs on and then constructing a nuclear potential energy curve, \(V_{r}(Q)\), via Equation \(\ref{12}\). The next step involves solving the nuclear Schrödinger equation, Equation \(\ref{11}\), with the constructed potential energy curve for the nuclear wavefunction, and then uses Equation \(\ref{4}\) with only one expansion term to create a total wavefunction for the system. This occurs only within the adiabatic approximation where the wavefunction can be expressed as only one product of the adiabatic electronic and nuclear wavefunctions.

    Breakdown of the Born-Oppenheimer Approximation

    The essence of the Born-Oppenheimer approximation is that the nuclear motion occurs on only one electronic energy surface. When in the course of the nuclear motion, possibly in intramolecular vibrations or intermolecular reactions, the nuclear motion can’t be expressed as occurring in one electronic state throughout the motion, then the Born-Oppenheimer approximation breaks down. In other words, the right side of Equation \(\ref{6}\) can’t be approximated as zero and thus the nuclear motion couples different electronic states. This is often called vibronic coupling. Any of the terms on the ride side of Equation \(\ref{6}\) can be responsible for non-adiabatic coupling of electronic energy states. Certainly, for diatomics with large nuclei where spin-orbit coupling becomes appreciable, the \(SO_{kk’}\) term can become a significant coupling factor. The \( T^{''}_{kk'}\) term though is generally small for the same reason that the \( T^{''}_{kk}\) was considered negligible in the previous section. The term mostly responsible for non-adiabatic transitions is the term \( T^{'}_{kk'}\). This term is velocity dependent since it involves the \(\bigtriangledown_{R}\) acting on the nuclear wavefunction \(\mid \chi_{k} \rangle\). Hence when nuclear motion is larger, the probability of non-adiabatic transitions increases, and thus non-adiabatic transitions can be observed in many high-energy chemical reactions. Equally, when the nature of the electronic wavefunction changes significantly within a short distant, e.g. an avoided crossing, this term increases in magnitude. Considerable interest has been generated in studying the effects of non-adiabatic transitions within the recent past and their effects in activated barrier crossings.


    There exists a connection between the adiabatic electronic energy curves constructed by solving Equation \(\ref{5}\) and the nuclear potential energy curves, \(V_{R}(Q)\), used to solve for nuclear motion. This connection can be a complicated one depending on the validity of the Born-Oppenheimer and adiabatic approximations used to solve the system. When the adiabatic approximation can be used then this relationship can be expressed in Equation \(\ref{12}\). And under circumstances when the BO approximation is valid and the spin-orbit coupling can be ignored, the nuclear potential curve can then be further approximated by the electronic energy curve \(\epsilon_{k}(Q)\). Naturally, further additions to the molecular Hamiltonian will introduce extra corrections to the nuclear potential energy curve. An example of such is the LM2M2* nuclear potential curve used for the helium dimer, in which relativistic retardation effects are introduced to account for the finite time interactions between the electrostatic Coulomb forces of the electrons in the rather large diatomic helium.


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    Nuclear Potential Energy Curves is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Delmar Larsen.