Adiabatic, Born-Oppenheimer, and Non-adiabatic Approaches

Abstract

A detailed derivation of the adiabatic approximation and the Born-Oppenheimer approximation is presented, the difference between these two approximations is discussed and the circumstances under which the adiabatic approximation collapses are discussed. It is shown that the solution of the Schrödinger equation in the adiabatic approximation can be divided into one representing the motion of electrons in the field of fixed nuclei and another one representing the motion of nuclei in the potential generated by the presence of the electrons. The shapes of the potential energy curves generated by the electrons and the motion of the nuclei in these potentials are also analyzed. Finally, the state-of-the-art highly accurate calculations for diatomic molecules performed without the use of the Born-Oppenheimer approximation is presented.

Appendices

Appendix A

Detailed derivation of the Laplace operator in the new coordinate (7), (8), (9),and (10)

• First derivatives of the nuclear coordinates:

  • nucleus a

    $$ \displaystyle\begin{array}{rcl} \frac{\partial } {\partial X_{a}^{{\prime}}}& =& \frac{\partial X_{\mathrm{CM}}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial X_{\mathrm{CM}}}+\frac{\partial Y _{\mathrm{CM}}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial Y _{\mathrm{CM}}}+\frac{\partial Z_{\mathrm{CM}}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial Z_{\mathrm{CM}}}+ \frac{\partial R_{x}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial R_{x}}+ \frac{\partial R_{y}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial R_{y}}+ \frac{\partial R_{z}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial R_{z}}+ \\ & +& \sum _{i=1}^{N_{e} }\left ( \frac{\partial x_{i}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial x_{i}} + \frac{\partial y_{i}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial y_{i}} + \frac{\partial z_{i}} {\partial X_{a}^{{\prime}}} \frac{\partial } {\partial z_{i}}\right )\; =\; \frac{M_{a}} {M} \frac{\partial } {\partial X_{\mathrm{CM}}} + \frac{\partial } {\partial R_{x}} -\frac{1} {2}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}} {}\end{array}$$

    (128)

    analogously:

    $$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial Y _{a}^{{\prime}}}& =& \frac{M_{a}} {M} \frac{\partial } {\partial Y _{\mathrm{CM}}} + \frac{\partial } {\partial R_{y}} -\frac{1} {2}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial y_{i}}, \\ \frac{\partial } {\partial Z_{a}^{{\prime}}}& =& \frac{M_{a}} {M} \frac{\partial } {\partial Z_{\mathrm{CM}}} + \frac{\partial } {\partial R_{z}} -\frac{1} {2}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial z_{i}} {}\end{array}$$

    (129)

  • nucleus b - analogous:

    $$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial X_{b}^{{\prime}}}& =& =\; \frac{M_{a}} {M} \frac{\partial } {\partial X_{\mathrm{CM}}} - \frac{\partial } {\partial R_{x}} -\frac{1} {2}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}} {}\end{array}$$

    (130)

  and similarly for the nuclear coordinates $\tilde{y}_{b}$ and $\tilde{z}_{b}$

• First derivatives of the electron coordinates:

  $$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial x^{{\prime}}_{i}}& =& \frac{X_{\mathrm{CM}}} {\partial x^{{\prime}}_{i}} \frac{\partial } {X_{\mathrm{CM}}}+\frac{Y _{\mathrm{CM}}} {\partial x^{{\prime}}_{i}} \frac{\partial } {Y _{\mathrm{CM}}}+\frac{Z_{\mathrm{CM}}} {\partial x^{{\prime}}_{i}} \frac{\partial } {Z_{\mathrm{CM}}}+ \frac{R_{x}} {\partial x^{{\prime}}_{i}} \frac{\partial } {R_{x}} + \frac{R_{y}} {\partial x^{{\prime}}_{i}} \frac{\partial } {R_{y}} + \frac{R_{z}} {\partial x^{{\prime}}_{i}} \frac{\partial } {R_{z}} + \\ & & +\!\sum _{j=1}^{N_{e} }\left ( \frac{\partial x_{j}} {\partial x^{{\prime}}_{i}} \frac{\partial } {\partial x_{j}}+ \frac{\partial y_{j}} {\partial x^{{\prime}}_{i}} \frac{\partial } {\partial y_{j}}+ \frac{\partial z_{j}} {\partial x^{{\prime}}_{i}} \frac{\partial } {\partial z_{j}}\right )= \frac{m} {M} \frac{\partial } {\partial X_{\mathrm{CM}}}+\!\sum _{j=1}^{N_{e} }\left (\delta _{ij} \frac{\partial } {\partial x_{j}}\right )= \\ & =& \frac{m} {M} \frac{\partial } {\partial X_{\mathrm{CM}}} + \frac{\partial } {\partial x_{i}} {}\end{array}$$

  (131)

  and similarly for the electron coordinates y ^′ _i and z ^′ _i

• Second derivatives of the nuclear coordinates (we use the formula $\left [a+(b+c)\right ]^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ac$):

  • nucleus a

    $$\displaystyle\begin{array}{rcl} \frac{\partial ^{2}} {\partial X^{{\prime}}_{a}{}^{2}}& =& \!\left (\frac{M_{a}} {M} \frac{\partial } {\partial X_{\mathrm{CM}}}+ \frac{\partial } {\partial R_{x}}-\frac{1} {2}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}}\right )\left (\frac{M_{a}} {M} \frac{\partial } {\partial X_{\mathrm{CM}}}+ \frac{\partial } {\partial R_{x}}-\frac{1} {2}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}}\right )\!= \\ & =& \left (\frac{M_{a}} {M} \right )^{2} \frac{\partial ^{2}} {\partial X_{\mathrm{CM}}^{2}} + \frac{\partial ^{2}} {\partial R_{x}^{2}} + \frac{1} {4}\sum _{i=1}^{N_{e} }\left ( \frac{\partial } {\partial x_{i}}\right ) + \\ & & +2\frac{M_{a}} {M} \frac{\partial ^{2}} {\partial X_{\mathrm{CM}}\partial R_{x}} - \frac{\partial } {\partial X_{\mathrm{CM}}}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}} -\frac{M_{a}} {M} \frac{\partial } {\partial R_{x}}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}} {}\end{array}$$

    (132)

  • nucleus b

    $$\displaystyle\begin{array}{rcl} \frac{\partial ^{2}} {\partial X^{{\prime}}_{b}{}^{2}}& =& \left (\frac{M_{b}} {M} \frac{\partial } {\partial X_{\mathrm{CM}}}- \frac{\partial } {\partial R_{x}}-\frac{1} {2}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}}\right )\left (\frac{M_{b}} {M} \frac{\partial } {\partial X_{\mathrm{CM}}}- \frac{\partial } {\partial R_{x}}-\frac{1} {2}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}}\right )= \\ & =& \left (\frac{M_{b}} {M} \right )^{2} \frac{\partial ^{2}} {\partial X_{\mathrm{CM}}^{2}} + \frac{\partial ^{2}} {\partial R_{x}^{2}} + \frac{1} {4}\sum _{i=1}^{N_{e} }\left ( \frac{\partial } {\partial x_{i}}\right ) + \\ & & -2\frac{M_{b}} {M} \frac{\partial ^{2}} {\partial X_{\mathrm{CM}}\partial R_{x}}+ \frac{\partial } {\partial X_{\mathrm{CM}}}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}}-\frac{M_{b}} {M} \frac{\partial } {\partial R_{x}}\sum _{i=1}^{N_{e} } \frac{\partial } {\partial x_{i}} {}\end{array}$$

    (133)

• Second derivatives of the electron coordinates:

  $$\displaystyle\begin{array}{rcl} \frac{\partial ^{2}} {\partial x^{{\prime}}_{i}{}^{2}}& =& \left ( \frac{m} {M} \frac{\partial } {\partial X_{\mathrm{CM}}} + \frac{\partial } {\partial x_{i}}\right )\left ( \frac{m} {M} \frac{\partial } {\partial X_{\mathrm{CM}}} + \frac{\partial } {\partial x_{i}}\right ) \\ & =& \left ( \frac{m} {M}\right )^{2} \frac{\partial ^{2}} {\partial X_{\mathrm{CM}}^{2}} + \frac{\partial ^{2}} {\partial x_{i}^{2}} + 2 \frac{m} {M} \frac{\partial ^{2}} {\partial X_{\mathrm{CM}}\partial x_{i}} {}\end{array}$$

  (134)

  Similarly, the remaining coordinates.

Appendix B

Flat rotor – a system of two particles with masses m ₁ and m ₂ at a constant distance R from each other. In this system there are no external forces, but it can rotate around its center of mass. Hamiltonian for this system has a form:

$$\displaystyle{ \hat{H}= - \frac{1} {2m_{1}}\nabla _{\mathbf{r}_{1}}^{2} - \frac{1} {2m_{2}}\nabla _{\mathbf{r}_{2}}^{2}, }$$

(135)

where r ₁ and r ₂ are vectors that describe the positions of the two bodies in the laboratory reference frame. Alternatively one can describe the same system, in the center-of-mass reference frame. The coordinates of the center of mass in the laboratory frame are given by

$$\displaystyle{ \mathbf{R} = \frac{m_{1}\mathbf{r}_{1} + m_{2}\mathbf{r}_{2}} {m_{1} + m_{2}} \; =\; \left [X,Y,Z\right ] }$$

(136)

If one wants to express the Hamiltonian in the center-of-mass coordinates, one has to express the second derivatives in the new variables. Proceeding as in Appendix A one gets

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial x_{1}}& =& \frac{\partial X} {\partial x_{1}}\; \frac{\partial } {\partial X} + \frac{\partial Y } {\partial x_{1}}\; \frac{\partial } {\partial Y } + \frac{\partial Z} {\partial x_{1}}\; \frac{\partial } {\partial Z} + \frac{\partial x} {\partial x_{1}}\; \frac{\partial } {\partial x} + \frac{\partial y} {\partial x_{1}}\; \frac{\partial } {\partial y} + \frac{\partial z} {\partial x_{1}}\; \frac{\partial } {\partial z} \\ & =& \frac{\partial X} {\partial x_{1}}\; \frac{\partial } {\partial X} + \frac{\partial x} {\partial x_{1}}\; \frac{\partial } {\partial x} \\ & =& \frac{m_{1}} {m_{1} + m_{2}}\; \frac{\partial } {\partial X} + \frac{\partial } {\partial x}, {}\end{array}$$

(137)

$$\displaystyle\begin{array}{rcl} \frac{\partial } {\partial x_{2}}& =& \frac{\partial X} {\partial x_{2}}\; \frac{\partial } {\partial X} + \frac{\partial Y } {\partial x_{2}}\; \frac{\partial } {\partial Y } + \frac{\partial Z} {\partial x_{2}}\; \frac{\partial } {\partial Z} + \frac{\partial x} {\partial x_{2}}\; \frac{\partial } {\partial x} + \frac{\partial y} {\partial x_{2}}\; \frac{\partial } {\partial y} + \frac{\partial z} {\partial x_{2}}\; \frac{\partial } {\partial z} \\ & =& \frac{\partial X} {\partial x_{2}}\; \frac{\partial } {\partial X} + \frac{\partial x} {\partial x_{2}}\; \frac{\partial } {\partial x} \\ & =& \frac{m_{2}} {m_{1} + m_{2}}\; \frac{\partial } {\partial X} - \frac{\partial } {\partial x}. {}\end{array}$$

(138)

Similarly, for the remaining coordinates. The second derivatives are given by

$$\displaystyle\begin{array}{rcl} \frac{\partial ^{2}} {\partial x_{1}^{2}}& =& \left ( \frac{m_{1}} {m_{1} + m_{2}}\; \frac{\partial } {\partial X} + \frac{\partial } {\partial x}\right ) \cdot \left ( \frac{m_{1}} {m_{1} + m_{2}}\; \frac{\partial } {\partial X} + \frac{\partial } {\partial x}\right ) \\ & =& \left ( \frac{m_{1}} {m_{1} + m_{2}}\right )^{2} \frac{\partial ^{2}} {\partial X^{2}} + \frac{2m_{1}} {m_{1} + m_{2}}\; \frac{\partial ^{2}} {\partial X\;\partial x} + \frac{\partial ^{2}} {\partial x^{2}}{}\end{array}$$

(139)

$$\displaystyle\begin{array}{rcl} \frac{\partial ^{2}} {\partial x_{2}^{2}}& =& \left ( \frac{m_{2}} {m_{1} + m_{2}}\; \frac{\partial } {\partial X} - \frac{\partial } {\partial x}\right ) \cdot \left ( \frac{m_{2}} {m_{1} + m_{2}}\; \frac{\partial } {\partial X} - \frac{\partial } {\partial x}\right ) \\ & =& \left ( \frac{m_{2}} {m_{1} + m_{2}}\right )^{2} \frac{\partial ^{2}} {\partial X^{2}} - \frac{2m_{2}} {m_{1} + m_{2}}\; \frac{\partial ^{2}} {\partial X\;\partial x} + \frac{\partial ^{2}} {\partial x^{2}}{}\end{array}$$

(140)

Thus, after substituting the second derivatives, one obtains the Hamiltonian operator (135) in the new variables

$$\displaystyle\begin{array}{rcl} \hat{H}& =& -\frac{1} {2}\left [ \frac{m_{1}} {(m_{1} + m_{2})^{2}} + \frac{m_{2}} {(m_{1} + m_{2})^{2}}\right ]\left ( \frac{\partial ^{2}} {\partial X^{2}} + \frac{\partial ^{2}} {\partial Y ^{2}} + \frac{\partial ^{2}} {\partial Z^{2}}\right ) \\ & -& \frac{1} {2}\left [ \frac{1} {m_{1}} + \frac{1} {m_{2}}\right ]\left ( \frac{\partial ^{2}} {\partial x^{2}} + \frac{\partial ^{2}} {\partial y^{2}} + \frac{\partial ^{2}} {\partial z^{2}}\right ) \\ & =& - \frac{1} {2M}\;\Delta _{\mathbf{R}} -\frac{1} {2\mu }\;\Delta _{\mathbf{R}} \equiv \hat{H} _{\mathbf{R}} + \hat{H} _{\mathbf{R}}, {}\end{array}$$

(141)

where

$$\displaystyle\begin{array}{rcl} M& =& m_{1} + m_{2},\qquad \frac{1} {\mu } = \frac{1} {m_{1}} + \frac{1} {m_{2}},\qquad \Delta _{\mathbf{R}} = \frac{\partial ^{2}} {\partial X^{2}} + \frac{\partial ^{2}} {\partial Y ^{2}} + \frac{\partial ^{2}} {\partial Z^{2}}, \\ \Delta _{\mathbf{R}}& =& \frac{\partial ^{2}} {\partial x^{2}} + \frac{\partial ^{2}} {\partial y^{2}} + \frac{\partial ^{2}} {\partial z^{2}}, {}\end{array}$$

(142)

and μ is the reduced mass. For the two particles moving freely in the space, that is assuming that V = 0, the eigenvalue equation of (141) can be written as $\hat{H}_{\mathbf{R}}$ and $\hat{H}_{\mathbf{R}}$:

$$\displaystyle{ (\hat{H} _{\mathbf{R}} + \hat{H} _{\mathbf{R}})\psi (\mathbf{R},\mathbf{r}) = E\psi (\mathbf{R},\mathbf{r}) }$$

(143)

for

$$\displaystyle{\psi (\mathbf{R},\;\mathbf{r}) =\psi _{\mathbf{R}}(\mathbf{R})\;\psi _{\mathbf{R}}(\mathbf{r}),}$$

where

$$\displaystyle{\hat{H} _{\mathbf{R}}\;\psi _{\mathbf{R}}(\mathbf{R}) = E_{\mathbf{R}}\;\psi _{\mathbf{R}}(\mathbf{R})\qquad \mbox{ and}\qquad \hat{H} _{\mathbf{R}}\;\psi _{\mathbf{R}}(\mathbf{r}) = E_{\mathbf{R}}\;\psi _{\mathbf{R}}(\mathbf{r}).}$$

We focus only on the relative motion Hamiltonian $\hat{H}_{\mathbf{R}}$. The eigenfunction ψ _R (r) of this Hamiltonian depends on three variables. However, due to the fact that r is constant (r ² = x ² + y ² + z ² = const), only two angular variables are independent. In the spherical coordinates, with variables r, θ, ϕ, where r = const:

$$\displaystyle\begin{array}{rcl} x& =& x(r,\theta,\phi ) = r\sin \theta \cos \phi \, {}\\ y& =& y(r,\theta,\phi ) = r\sin \theta \sin \phi \, {}\\ z& =& z(r,\theta,\phi ) = r\cos \theta. {}\\ \end{array}$$

the Laplacian can be written as

$$\displaystyle{\Delta _{\mathbf{R}} = \frac{1} {r^{2}}\left [ \frac{\partial } {\partial r}\left (r^{2} \frac{\partial } {\partial r}\right ) + \frac{1} {\sin \theta } \frac{\partial } {\partial \theta }\left (\sin \theta \frac{\partial } {\partial \theta }\right ) + \frac{1} {\sin ^{2}\theta } \frac{\partial ^{2}} {\partial \phi ^{2}}\right ].}$$

Consequently, the rigid rotor Hamiltonian becomes

$$\displaystyle{\hat{H}= -\frac{1} {2I}\left [\frac{1} {\sin \theta } \frac{\partial } {\partial \theta }\left (\sin \theta \frac{\partial } {\partial \theta }\right ) + \frac{1} {\sin ^{2}\theta } \frac{\partial ^{2}} {\partial \phi ^{2}}\right ],}$$

where

$$\displaystyle{I =\mu R^{2}}$$

is the moment of inertia. The eigenvalue problem of the rigid rotor is, thus, reduced to the equation

$$\displaystyle{-\frac{1} {2I}\left [\frac{1} {\sin \theta } \frac{\partial } {\partial \theta }\left (\sin \theta \frac{\partial } {\partial \theta }\right ) + \frac{1} {\sin ^{2}\theta } \frac{\partial ^{2}} {\partial \phi ^{2}}\right ]Y (\theta,\phi ) = EY (\theta,\phi ),}$$

analogous to the problem of the angular momentum

$$\displaystyle{-1\left [\frac{1} {\sin \theta } \frac{\partial } {\partial \theta }\left (\sin \theta \frac{\partial } {\partial \theta }\right ) + \frac{1} {\sin ^{2}\theta } \frac{\partial ^{2}} {\partial \phi ^{2}}\right ]Y (\theta,\phi ) = 1l(l + 1)Y (\theta,\phi ).}$$

Thus, we can immediately conclude that the rigid rotor eigenfunctions are equal to the spherical harmonics

$$\displaystyle{Y _{J}^{M}(\theta,\phi ) = \frac{1} {\sqrt{2\pi }}N_{J,\vert M\vert }P_{J}^{\vert M\vert }(\cos \theta )e^{iM\phi },}$$

where

$$\displaystyle{P_{l}^{m}(x) = (1 - x^{2})^{m/2} \frac{d^{m}} {dx^{m}}P_{l}(x)}$$

is the associate Legendre polynomial. The eigenvalues are given by

$$\displaystyle{E_{J} = \frac{1} {2I}J(J + 1),}$$

where 1∕2I is the rotational constant.