Vibrational normal coordinates
The potential energy is approximated by a second-order Taylor expansion around the stationary geometry x0.
V(x)~ V(xo ) + (x - xo) + "2 (x - xo )t IdV}* - Xo) (1637)
The energy for the expansion point, V(x0), may be chosen as zero, and the first derivative is zero since x0 is a stationary point.
Here F is a 3Natom x 3Natom (force constant) matrix containing the second derivatives of the energy with respect to the coordinates. The nuclear Schrödinger equation for an Natom system is given by eq. (16.39).
3 Watom 1 d2
Eq. (16.39) is first transformed to mass-dependent coordinates by a G matrix containing the inverse square root of atomic masses (note that atomic, not nuclear, masses are used, which is in line with the Born-Oppenheimer approximation that the electrons follow the nuclei).
dyf mi dxf
3Watpm/ d 2
A unitary transformation is then introduced that diagonalizes the F G (entrywise product, eq. (16.10)) matrix, yielding eigenvalues ei and eigenvectors qi. The kinetic energy operator is still diagonal in these coordinates.
nuc nuc nuc
£ (|1 qt(u(F-G)ut=enuc^r 
£ (|1 qt(u(F-G)ut=enuc^r 
In the q-coordinate system, the vibrational normal coordinates, the 3Natom-dimensional Schrödinger equation can be separated into 3Natom one-dimensional Schrödinger equations, which are just in the form of a standard harmonic oscillator, with the solutions being Hermite polynomials in the q-coordinates. The eigenvectors of the FG matrix are the (mass-weighted) vibrational normal coordinates, and the eigenvalues ei are related to the vibrational frequencies as shown in eq. (16.42) (analogous to
When this procedure is carried out in Cartesian coordinates, there should be six (five for a linear molecule) eigenvalues of the F G matrix being exactly zero, corresponding to the translational and rotational modes. In real calculations, however, these values are not exactly zero. The three translational modes usually have "frequencies" very close to zero, typically less than 0.01cm-1. The deviation from zero is due to the fact that numerical operations are only carried out with a finite precision, and the accumulations of errors will typical give inaccuracies in v of this magnitude. The residual "frequencies" for the rotational modes, however, may often be as large as 1050 cm-1. This is due to the fact that the geometry cannot be optimized to a gradient of exactly zero, again due to numerical considerations. Typically, the geometry optimization is considered converged if the root mean square (RMS) gradient is less than ~10-4-10-5au, corresponding to the energy being converged to ~10-5-10-6au. The residual gradient shows up as vibrational frequencies for the rotations of the above magnitude.
If there are real frequencies of the same magnitude as the "rotational frequencies", mixing may occur and result in inaccurate values for the "true" vibrations. For this reason, the translational and rotational degrees of freedom are normally removed by projection (Section 16.4) from the force constant matrix before diagonalization.
If the stationary point is a minimum on the energy surface, the eigenvalues of the F and F G matrices are all positive. If, however, the stationary point is a transition state (TS), one (and only one) of the eigenvalues is negative. This corresponds to the energy being a maximum in one direction and a minimum in all other directions. The "frequency" for the "vibration" along the eigenvector with a negative eigenvalue will formally be imaginary, as it is the square root of a negative number (eq. (16.42)). The corresponding eigenvector is the direction leading downhill from the TS towards the reactant and product. At the TS, the eigenvector for the imaginary frequency is the eq. (13.31)).
reaction coordinate. The whole reaction path may be calculated by sliding downhill to each side from the TS. This can be performed by taking a small step along the TS eigenvector, calculating the gradient and taking a small step in the negative gradient direction. The negative of the gradient always points downhill, and by taking a sufficiently large number of such steps an energy minimum is eventually reached. This is equivalent to a steepest descent minimization, but more efficient methods are available (see Section 12.8 for details). The reaction path in mass-weighted coordinates is called the Intrinsic Reaction Coordinate (IRC).
The vibrational Hamiltonian is completely separable within the harmonic approximation, with the vibrational energy being a sum of individual energy terms and the nuclear wave function being a product of harmonic oscillator functions (Hermite polynomial in the normal coordinates). When anharmonic terms are included in the potential, the Hamiltonian is no longer separable, and the resulting nuclear Schrödinger equation can be solved by techniques completely analogous to those used for solving the electronic problem. The vibrational SCF method is analogous to the electronic Hartree-Fock method, with the nuclear harmonic oscillator functions playing the same role as the orbitals in electronic structure theory. Corrections beyond the mean-field approximation can be added by configuration interaction, perturbation theory or coupled cluster methods.2
It should be noted that the force constant matrix can be calculated at any geometry, but the transformation to normal coordinates is only valid at a stationary point, i.e. where the first derivative is zero. At a non-stationary geometry, a set 3Natom - 7 generalized frequencies may be defined by removing the gradient direction from the force constant matrix (for example by projection techniques, Section 16.4) before transformation to normal coordinates.
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