The Potential Energy Surface in Molecular Quantum Mechanics

Abstract

The idea of a Potential Energy Surface (PES) forms the basis of almost all accounts of the mechanisms of chemical reactions, and much of theoretical molecular spectroscopy. It is assumed that, in principle, the PES can be calculated by means of clamped-nuclei electronic structure calculations based upon the Schrödinger Coulomb Hamiltonian. This article is devoted to a discussion of the origin of the idea, its development in the context of the Old Quantum Theory, and its present status in the quantum mechanics of molecules. It is argued that its present status must be regarded as uncertain.

Appendix

In this appendix we give an heuristic account of the mathematical notion of the direct integral of Hilbert spaces, and then study a simple model problem to illustrate the general ideas discussed in the paper.

Consider a self-adjoint operator T that depends on a parameter X, so T=T(X). The parameter X=−∞≤X≤+∞, covers the whole real line , and T(X) is assumed to be defined for all X. T(X) is an operator on a Hilbert space, which is denoted ; its eigenfunctions {ϕ} defined by

$$T(X) \phi(X)_k=\lambda_k(X) \phi(X)_k, $$

form a complete orthogonal set for the space . The scalar product is

$$ \bigl\langle\phi(X)_{k} \big|\phi(X)_{j} \bigr \rangle_{X}=\int\phi(X,x)_{k}^{*}\phi (X,x)_{j} dx=f(X)_{kj} \equiv f(X) \delta_{kj},\quad f(X)<\infty $$

(1.51)

T may have discrete eigenvalues below a continuous spectrum that starts at Λ, that is

$$\sigma(X)=\sigma\bigl(T(X)\bigr)=\bigl[\lambda_0(X), \lambda_1(X),\ldots ,\lambda_k(X) \bigr) \bigcup \bigl[ \varLambda(X),\infty \bigr). $$

Now let’s introduce a ‘big’ Hilbert space as a direct integral over the {},

(1.52)

and correspondingly the operator acting on defined by

The scalar product on the big space is defined explicitly in terms of (1.51) by

(1.53)

In (1.51) one can always chose the functions {ϕ _k } to be orthonormalized independently of X,

$$f(X)=1. $$

However this choice is not consistent with (1.53), which requires f(X) to decrease sufficiently fast as |X|→∞ for the integral to exist. The mathematical motivation for this construction is this: the cartesian product of the spaces {},

is a field of Hilbert spaces over which has a natural vector space structure. In modern geometric language, the Hilbert space is a fibre over a point X in a fibre bundle; is the vector space of sections of this bundle. The subspace of consisting of square integrable sections is the direct integral (1.52). The direct integral is the generalization to the continuous infinite dimensional case of the notion of the direct sum of finite dimensional vector spaces.

Purely as a heuristic explanation suppose initially that the parameter X has only two discrete values {X ₁,X ₂}. There are then two eigenvalue equations to consider, and two associated Hilbert spaces. In the direct sum space we have

The eigenfunctions of are then two-dimensional column vectors, and so

The spectrum of is the union of the spectra of T(X ₁) and T(X ₂). This discussion is trivially extended to n points {X _k :k=1,…,n}, with the spectrum of given by the union of the n operators T(X ₁),…,T(X _n ). The limit n→∞ is not trivial since it brings in important notions from topology and integration (measure theory) which we gloss over [94]. When these are taken into consideration however the result is that the spectrum σ of is purely continuous since its direct integral representation implies that its spectrum is the union of the spectra of the infinite set of T(X) operators,

(1.54)

where L ₀ is the minimum value of λ ₀(X). The eigenvalue equation for is,

with Λ a continuous index for the {Φ}. Even if T(X) is self-adjoint, it doesn’t follow that its direct integral is self-adjoint; that depends on specifics and has to be investigated. So the {Φ} cannot be assumed to be complete.

In the application of this mathematics to the Born-Oppenheimer approximation, the role of x is taken by the electronic coordinates t ^e, and X is to be identified with definite choices of the nuclear coordinates b. If there are M nuclei the parameters b are elements of . The operator T(X) is the clamped-nuclei Hamiltonian $\hat{\mathsf{K}}(\mathbf{b},\hat{\mathbf{t}}^{\mathrm{e}})_{o}=\hat{\mathsf{K}}_{o}$. With the conventional normalization of clamped-nuclei electronic eigenfunctions independent of the nuclear positions b, the formal eigenvectors, (1.36), [56] of $\hat{\mathsf{H}}^{\mathrm{elec}}$ do not belong to the Hilbert space ; this simply reflects the loose use of the Dirac delta function for the position operator eigenfunctions.

We now consider a concrete model consisting of coupled harmonic oscillators with two degrees of freedom; we try to mimic the steps taken in the usual Born-Oppenheimer discussion. Consider the following Hamiltonian where κ and a are dimensionless constants²⁰

$$ \hat{\mathsf{H}}=\hat{\mathsf{p}}^2+\kappa^4\hat{ \mathsf{P}}^2+\hat{\mathsf{x}}^2+\hat{ \mathsf{X}}^2+a\hat{\mathsf{x}}\hat{\mathsf{X}}. $$

(1.55)

The only non-zero commutators of the operators are

$$[\hat{\mathsf{x}},\hat{\mathsf{p}}]=i ,\qquad [\hat{\mathsf{X}},\hat{ \mathsf{P}}]=i. $$

Following the conventional discussion of electron-nuclear separation outlined in Sects. 1.3.1, 1.3.2, define

$$\begin{aligned} \hat{\mathsf{H}}_o =&\hat{\mathsf{p}}^2+\hat{ \mathsf{x}}^2+a\hat{\mathsf{x}}\hat{\mathsf{X}}+\hat{ \mathsf{X}}^2 \end{aligned}$$

(1.56)

$$\begin{aligned} \hat{\mathsf{H}}_1 =&\hat{\mathsf{P}}^2 \end{aligned}$$

(1.57)

so that

$$ \hat{\mathsf{H}}=\hat{\mathsf{H}}_o+\kappa^4 \hat{ \mathsf{H}}_1 $$

(1.58)

with Schrödinger equation

$$ (\hat{\mathsf{H}}-E )\varPsi=0. $$

(1.59)

We note that

$$ [\hat{\mathsf{H}}_o,\hat{\mathsf{H}}_1] \neq 0 $$

(1.60)

so the two parts cannot be simultaneously diagonalized. A principal axis transformation of the whole expression $\hat{\mathsf{H}}$ brings it to separable form, but we do not need to pursue explicitly the full solution here.

On the other hand

$$ [\hat{\mathsf{H}}_o,\hat{\mathsf{X}}]=0 $$

(1.61)

so these two operators may be simultaneously diagonalized, and consider $\hat{\mathsf{H}}_{o}$ at a definite eigenvalue of $\hat{\mathsf{X}}$, say X

$$ \hat{\mathsf{K}}_o=\hat{\mathsf{p}}^2+\hat{ \mathsf{x}}^2+a\hat{\mathsf{x}} {X}+{X}^2. $$

(1.62)

This is the Hamiltonian (in the variables $\hat{\mathsf{x}},\hat{\mathsf{p}}$) of a displaced oscillator in which X is a (c-number) parameter, with Schrödinger equation in position representation

$$ \hat{\mathsf{K}}_o \varphi(x,X)_n=E_n(X) \varphi(x,X)_n $$

(1.63)

$\hat{\mathsf{K}}_{o}$ is the analogue in this model of the ‘clamped-nuclei’ electronic Hamiltonian.

The solution is immediate; we make a unitary transformation by introducing a displaced coordinate involving X

$$\begin{aligned} \hat{\mathsf{x}}' =&\hat{\mathsf{x}}-\frac{1}{2}aX,\qquad \hat{\mathsf{p}}'=\hat{\mathsf{p}} \end{aligned}$$

(1.64)

$$\begin{aligned} \hat{\mathsf{U}} =&e^{iaX\hat{\mathsf{p}}/2},\qquad \hat{\mathsf{K}}_{o}^{\prime}= \hat{\mathsf{U}}^{-1}\hat{\mathsf{K}}_0 \hat{\mathsf{U}} \end{aligned}$$

(1.65)

so that the transformed $\hat{\mathsf{K}}_{o}$ in the new variables is

$$\hat{\mathsf{K}}'_{o}=\hat{\mathsf{p}}'{}^{2}+ \hat{\mathsf{x}}'{}^{2} +\biggl(1-\frac{a^2}{4} \biggr)X^2 $$

which has oscillator eigenvalues and eigenfunctions

$$ E_n(X)=E_n^0+\biggl(1-\frac{a^2}{4} \biggr)X^2,\qquad \varphi\bigl(x'\bigr)_n $$

(1.66)

where $x'=x-\frac{aX}{2}$, the {φ _n } are the usual harmonic oscillator eigenfunctions and $E_{n}^{0}$ is the energy of the free oscillator $2(n+\frac{1}{2})$. For fixed n the spectrum σ(X) is discrete and, as a function of the X parameters, would be conventionally interpreted as a ‘potential energy curve’. As far as (1.58) is concerned, $\hat{\mathsf{K}}_{o}$, evaluated at some point X ₀ cannot be regarded as an ‘approximation’ to $\hat{\mathsf{H}}_{o}$, since obviously

$$\bigl[\hat{\mathsf{K}}_{o}(X_0),\hat{ \mathsf{H}}_1\bigr]=0 $$

so they can be simultaneously diagonalized, and $\hat{\mathsf{H}}_{1}$ has purely continuous spectrum (free motion). So we have to consider X in the full problem in its operator form, $\hat{\mathsf{X}}$.

We make the same unitary transformation of the $\hat{\mathsf{x}},\hat{\mathsf{p}}$ variables in $\hat{\mathsf{H}}_{o}$ as before, and it is still brought to diagonal form; however $\hat{\mathsf{H}}_{1}$ will be modified because $\hat{\mathsf{P}}$ is also translated by the operator $\hat{\mathsf{U}}$ in (1.65) that generates the coordinate displacement (cf. (1.60)); thus

$$\hat{\mathsf{U}}^{-1} \hat{\mathsf{P}} \hat{\mathsf{U}}=\hat{ \mathsf{P}}+\frac{1}{2}a\hat{\mathsf{p}} $$

so the transform of $\hat{\mathsf{H}}_{1}$ contains linear and quadratic terms in $\hat{\mathsf{p}}$.

Nevertheless (1.61) is still valid, and formally we may write $\hat{\mathsf{H}}_{0}$ as a direct integral

$$ \hat{\mathsf{H}}_o=\int^{\oplus}_{X} \hat{\mathsf{K}}(\hat{\mathsf{x}},\hat{\mathsf{p}},X)_o dX. $$

(1.67)

The Schrödinger equation for $\hat{\mathsf{H}}_{o}$ in position representation is now an equation involving functions of x and X

$$ \hat{\mathsf{H}}_o \varPhi_{\varepsilon}\bigl(x',X \bigr)=\varepsilon \varPhi_{\varepsilon}\bigl(x',X\bigr). $$

(1.68)

Just as before (see (1.54)) the direct integral decomposition (1.67) implies that the spectrum is purely continuous, explicitly

$$ \sigma(\hat{\mathsf{H}}_o)=\bigcup_{X} \sigma(X)= [1,\infty ). $$

(1.69)

ε in (1.68) takes all positive values ≥1, where 1 is the minimum value of the oscillator eigenvalue $2(n+\frac{1}{2})$. The associated continuum eigenfunctions {Φ} may formally be written as products of oscillator eigenfunctions (in x′), and delta functions (in X). They don’t lie in Hilbert space of course and one needs the Gel’fand construction of a rigged space to make sense of the formal calculation [95]. If one returns to the x variable, the {φ _n } are functions of x and X, since x′=x′(X).

One can’t do anything very useful with the direct integral expression (1.67) for $\hat{\mathsf{H}}_{o}$ apart from adding it onto the κ ⁴ term, which just returns us to the full problem. The full wavefunction in (1.59) can be expanded as

(1.70)

$$\begin{aligned} =&\sum_n c(X)_n \varphi \bigl(x'(X) \bigr)_n \end{aligned}$$

(1.71)

which obviously leads towards a variational approach [56]; such expansions rely on the completeness of the states employed. In this simple problem there is no difficulty, but as noted earlier, in realistic Coulomb systems it is much less clear that a complete set of states is available.

However that may be, let us rehearse again the argument due to Born summarized in Sect. 1.3.2. We substitute (1.71) in (1.59), left multiply by $\varphi_{m}^{*}$ and integrate out the x′ variables to leave an equation for the coefficients {c(X) _n },

$$\begin{aligned} &\int dx' \varphi \bigl(x'(X) \bigr)_{m}^{*} \bigl[\hat{\mathsf{H}}_o+\kappa^4 \hat{ \mathsf{H}}_1 \bigr] \sum_{n}c(X)_n \varphi \bigl(x'(X) \bigr)_n \\ &\quad= E \int dx' \varphi \bigl(x'(X) \bigr)_{m}^{*} \sum_{n}c(X)_n \varphi \bigl(x'(X) \bigr)_n. \end{aligned}$$

(1.72)

At this point in the conventional account, $\hat{\mathsf{H}}_{o}$ is replaced by $\hat{\mathsf{K}}_{o}$, (1.62), and then the action of $\hat{\mathsf{K}}_{o}$ on the functions {φ} in (1.72) can be evaluated using (1.63) in the well-known way,

$$\hat{\mathsf{H}}_o \varphi \bigl(x'(X) \bigr)_n \rightarrow \hat{\mathsf{K}}_o \varphi \bigl(x'(X) \bigr)_n=\varepsilon(X)_n \varphi \bigl(x'(X) \bigr)_n. $$

From the foregoing discussion it is clear that the substitution of $\hat{\mathsf{H}}_{o}$ by $\hat{\mathsf{K}}_{o}$ makes a qualitative change in the theory. This change does seem to be the ‘right’ thing to do, but so far there is no explanation as to why this is so.