The harmonic oscillator

The next stage in our development of the internal dynamics of a diatomic molecule is to recognise that the molecule is not rigid. The atoms with point masses m and m2 are at distances R and R2 from the centre-of-mass at any given instant, but actually move with respect to each other, with displacements from each equilibrium position given by 

The simplest definition of the vibrational coordinate, q, for a diatomic molecule is, in fact, 

The final result in (6.169) follows from the fact that the momentum p is equal to p(Aq/At). 

Equation (6.169) gives the kinetic energy, but we have still to obtain a classical expression for the potential energy. A harmonic oscillator can be defined as a mass point m which is acted upon by a force F proportional to the distance x from the equilibrium position and directed towards the equilibrium position. Since force equals mass times acceleration, we have 

In terms of the vibrational coordinate, we have by analogy with (6.173), 

X = xo sin(27TVosci + p), where the vibrational frequency Vosc is given by 

The potential well that restrains a chemical bond near its mean length is described reasonably well by the quadratic expression 

A classical particle of mass w in a harmonic potential well oscillates about its equilibrium position with a frequency given by 

Such a system is called a harmonic oscillator. Equation (2.28) also applies to the frequency of the classical vibrations of a pair of bonded atoms if we replace m by the reduced mass of the pair [m = mim2l(mi + m2), where mi and m2 are the masses of the individual atoms]. The angular frequency co is 2md = klmY.  

The sum of the kinetic and potential energies of a classical harmonic oscillator is 

Here is a normalization factor, // ( ) is a Hermite polynomial, and m is a dimensionless positional coordinate obtained by dividing the Cartesian coordinate 

The Harmonic Oscillator Linear Differential Equations with Constant Coefficients 

Replacement of F by md zldt according to Newton s second law gives the equation of motion for our harmonic oscillator  

It is called an ordinary differential equation because it contains only ordinary derivatives as opposed to partial derivatives. 

It is linear, which means that the dependent variable z and its derivatives enter only to the first power. 

It is homogeneous, which means that there are no terms that do not contain z. 

The harmonic oscillator is one of the most important elementary models in mechanics, and is especially relevant in chemistry in connection with the vibrations of molecules. 

A harmonic oscillator is a system where the force on the particle varies with its position according to 

Equation 3.9 describes the particle oscillating regularly about a mean position (x = 0), at a rate proportional to ft). This rate is often expressed in terms of the frequency of oscillation, V, which specifies the number of complete cycles per unit time. It is given by 

A is the amplitude of oscillation. An important property of the harmonic oscillator is that its frequency does not depend on this amplitude. 

From our results we can obtain expressions for the kinetic and potential energy of the oscillator. Thus 

The harmonic oscillator is an important model problem in chemical systems to describe the oscillatory (vibrational) motion along the bonds between the atoms in a molecule. In this model, the bond is viewed as a spring with a force constant of k. 

Taking the derivative of the Hamiltonian (Equation 1-7) with respect to position and applying Equation 1-5 yields  

The second differential equation yields a trivial result  

The solution to this differential equation is well known. One solution is given below. 

This relation says that there is an uncertainty in the energy of a particle and an uncertainty in the time at which the particle passes a given point in space the product of these uncertainties must equal or exceed h/4n. 

Heisenberg s development of quantum mechanics began with the uncertainty relations and led to the quantum mechanical equation. Schrodinger s treatment began with the wave equation and, as we have seen, we can argue from that to the uncertainty relations. 

Consider a particle moving in one dimension, along the x-axis, and bound to the origin (x = 0) by a Hooke s law restoring force, —kx. Newton s law, ma = F, then reads m d x/dt ) = —kx, or 

Equation (21.30) shows that the particle moves between — Xo and +Xo with a sinusoidal motion. Equation (21.31) shows that the velocity varies between —coxo and +coxo and is 90° out of phase with respect to the position. The total energy is 

Note that the energy of the classical oscillator depends only on the force constant, fc, and on the maximum displacement, Xo, which is an arbitrary quantity. The oscillator may have any total energy, depending on how large we make Xo. 

The quantum mechanical hamiltonian for a one-dimensional harmonic oscillator is given by 

As expected, the frequency increases with k (the stiffness of the bond) and decreases with /r. More commonly, though, we use vibration wavenumber co, rather than frequency, where 

Equation (1.69) shows the vibrational levels to be equally spaced, by hcco, and that the w = 0 level has an energy j hcco, known as the zero-point energy. This is the minimum energy 

Each point of intersection of an energy level with the curve corresponds to a classical turning point of a vibration where the velocity of the nuclei is zero and all the energy is in the form of potential energy. This is in contrast to the mid-point of each energy level where all the energy is kinetic energy. 

The wave functions ij/ resulting from solution of Equation (1.66) are 

A one-dimensional harmonic oscillator is a particle of mass m, subject to force force constant —kx, where the force constant A 0, and x is the displacement of the particle from its equilibrium position (.r = 0). This means the force pushes the particle 

The harmonic oscillator finger print it hao an infinite number of energy 

that the oscillator energy is never equal to zero. 

Charles Hermite was French mathematician (1822-1901), professor at the Sotbonne. The Hermite polynomials were defined half a century earlier by Pierre Laplace. 

The potential energy of a body moving under the action of a harmonic restraining force (Fig. 3.1) is y = l/2fcc, where A is the force constant. The quantum mechanical equation for the stationary states of a harmonic oscillator of mass mp is  

The algebra for the solution of this equation is long and complex, but the final result for the quantized energies turns out again in a simple and elegant form  

The classical vibrational Ifequency is denoted by v and the vibrational quantum number by v. 

A classical oscillator in its lowest energy state is at rest at x = 0. A quantum mechanical oscillator can never be constrained at x = 0 because the wavefunctions are non-zero over the whole range. Besides, the energy is not zero even when v = 0 the vibrational zero-point energy is E° = l/2hv. As for linear momentum, this zero-point energy is ultimately connected with the uncertainty principle. 

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Marquardt R and Quack M 1996 Radiative excitation of the harmonic oscillator with applications to stereomutation in chiral molecules Z. Rhys. D 36 229-37... 

The coordinates of interest to us in the following discussion are Qx and Qy, which describe the distortion of the molecular triangle from Dy, symmetry. In the harmonic-oscillator approximation, the factor in the vibrational wave... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

The hamionic oscillator (Fig. 4-1) is an idealized model of the simple mechanical system of a moving mass connected to a wall by a spring. Oirr interest is in ver y small masses (atoms). The harmonic oscillator might be used to model a hydrogen atom connected to a large molecule by a single bond. The large molecule is so... 

These are all empirical measurements, so the model of the harmonic oscillator, which is pur ely theoretical, becomes semiempirical when experimental information is put into it to see how it compares with molecular vibration as determined spectroscopically. In what follows, we shall refer to empirical molecular models such as MM, which draw heavily on empirical information, ab initio molecular models such as advanced MO calculations, which one strives to derive purely from theory without any infusion of empirical data, and semiempirical models such as PM3, which are in between (see later chapters). 

Equal spacing between energy levels is not unusual. In the case of the harmonic oscillator, it is the rule. 

The problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

Experimental. The vibrational spectrum of an ideal harmonic oscillator would consist of one line at frequency v corresponding to A = hv, where A is the distance between levels on the vertical energy axis in Fig. 10-la. In the harmonic oscillator, AE is the same for a transition from one energy level to an adjacent level. A selection rule An = 1, where n is the vibrational quantum number, requires that the transition be to an adjacent level. 

We don t know A vib but we can approximate it from the vibrational spacing of the bond vibrations in the harmonic oscillator approximation. 

The Morse oscillator model is often used to go beyond the harmonic oscillator approximation. In this model, the potential Ej(R) is expressed in terms of the bond dissociation energy Dg and a parameter a related to the second derivative k of Ej(R) at Rg k = ( d2Ej/dR2) = 2a2Dg as follows ... 

For very-high-accuracy ah initio calculations, the harmonic oscillator approximation may be the largest source of error. The harmonic oscillator frequencies... 

Figure 1.13 Plot of potential energy, V(r), against bond length, r, for the harmonic oscillator model for vibration is the equilibrium bond length. A few energy levels (for v = 0, 1, 2, 3 and 28) and the corresponding wave functions are shown A and B are the classical turning points on the wave function for w = 28...

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

We have seen in Section 1.3.6 how the vibrational energy levels of a diatomic molecule, treated in the harmonic oscillator approximation, are given by... 

However, unlike electrical anharmonicity, mechanical anharmonicity modifies the vibrational term values and wave functions. The harmonic oscillator term values of Equation (6.3) are modified to a power series in (u + ) ... 

One effect of the modification to the harmonic oscillator term values is that, unlike the case of the harmonic oscillator, cOg cannot be measured directly. Wavenumbers AO y2 for v+ ) — V transitions are given by... 

In an approximation which is analogous to that which we have used for a diatomic molecule, each of the vibrations of a polyatomic molecule can be regarded as harmonic. Quantum mechanical treatment in the harmonic oscillator approximation shows that the vibrational term values G(v ) associated with each normal vibration i, all taken to be nondegenerate, are given by... 

The vibrational term values for a polyatomic anharmonic oscillator with only nondegenerate vibrations are modified from the harmonic oscillator values of Equation (6.41) to... 

The simplest model for vibrational energy levels is the harmonic oscillator, which has allowed levels with energy... 

According to (2.29), dissipation reduces the spread of the harmonic oscillator making it smaller than the quantum uncertainty of the position of the undamped oscillator (de Broglie wavelength). Within exponential accuracy (2.27) agrees with the Caldeira-Leggett formula (2.26), and similar expressions may be obtained for more realistic potentials. 

The corresponding level broadening equals half. In fact is the diagonal kinetic coefficient characterizing the rate of phonon-assisted escape from the ground state [Ambegaokar 1987]. In harmonic approximation for the well the only nonzero matrix element is that with /= 1,K0 Q /> = <5o, where is the zero-point spread of the harmonic oscillator. For an anharmonic potential, other matrix elements contribute to (2.52). 

We proceed now to the calculation of B, following [Benderskii et al. 1992a]. The denominator in (4.11) (apart from normalization) is equal to the harmonic-oscillator partition function [2 sinh(ico+ )] The numerator is the product of the s satisfying an equation of the Shrodinger type... 

It is noteworthy that eq. (4.15a) is nothing but the linearized classical upside-down barrier equation of motion (8S/8x = 0) for the new coordinate x. Therefore, while x = 0 corresponds to the instanton, the nonzero solution to (4.15a) describes how the trajectory escapes from the instanton solution, when it deviates from it. The parameter X, referred to as the stability angle [Gutzwil-ler 1967 Rajaraman 1975], generalizes the harmonic-oscillator phase co, which would appear in (4.15), if CO, were a constant. The fact that X is real indicates the aforementioned instability of the instanton in two dimensions. Guessing that the determinant det( — -I- co, ) is a function of X only,... 

Rationalize nonzero zeio-point energies by reference to the harmonic oscillator model once again, and its energy ... 

Relate characterization of stationary points via the eigenvalues of the Hessian to the corresponding matrix under the harmonic oscillator problem. 

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