Eigenstates for large amplitude motions

9.8 Eigenstates for large amplitude motions

Energy eigenstates for large amplitude motions are needed for computing densities of states and partition functions. In the MultiWell Program Suite, the eigenstates are computed by solving the Schrödinger equation, as described here.

Theory

The Schrödinger equation for one-dimensional large-amplitude vibration separable from all other motions in the molecule is written as follows 58,59,69,70

& % + Vq () ' ! () q = E ! () q (1) ( & 2 $ qI eff $ q

where q is the vibrational coordinate, ħ is Planck’s constant, E and ψ are energy eigenvalue and eigenvector, respectively. V(q) is the vibrational potential energy function, and I eff is the effective mass, which is in general a function of coordinate q, but which, depending on V(q), is sometimes

constant. I 58,59,71-73 eff can be derived from the ro-vibrational G matrix, which is defined as:

=$ T (2) Y % &

where the I matrix is the 3×3 moment of inertia tensor, the Y matrix (N vib ×N vib ) is the pure vibration contribution, and the X matrix (3×N vib ) corresponds to the vibration-rotation interaction

(Coriolis) terms. Here, N vib is the number of vibration modes, which equals unity for a 1-D separable degree of freedom. All elements of I, X, and Y can be computed from the molecular structure as:

N atom

2 I kk =

& ( r ! $ r ! ) % () r ! k ' ; k = x, y, or z

(3a)

! = 1 N atom

I kk ' = " mrr !! k ! k ' ; k ≠ k’

X ij = * m ! & r ! % ' (4)

! = 1 ' $ q i &% (' $ q j & ( where the α index runs on the number of atoms (N atom ) in the molecule.

Y ij = ) m ! % % & (5)

For one-dimensional large-amplitude vibrations, the ro-vibrational G matrix is expressed as:

By inverting the right hand side of eq. (6), one obtains:

Finally, I -1

eff is obtained as: I eff =g 44 .

Note that in this work, central finite differences are used to compute the derivatives in Eqs. (4) and (5). Thus the steps between adjacent positions (e.g. dihedral angles) must be small enough so that the derivatives are sufficiently accurate. Acceptable step-sizes must be found by trial and error.

Hindered internal rotations

For hindered internal rotations, the large-amplitude motions are torsional changes in dihedral angles. The Schrödinger equation (1) for a 1-D torsion can be rewritten as follows 70 :

(8a) ) ' 2 % ! I eff ( ! % !

+ V ( !"! ( ( = E "! ( .

By defining the rotational constant B hr =

, we obtain:

2 I eff

B hr () # +V () # *# () =E *# () (8b)

where χ is the torsional (dihedral) angle (0≤χ<2π), V(χ) is the torsional potential energy function, and I eff is the effective reduced moment of inertia. Both I eff and B hr are in general functions of dihedral angle, but are constants for a rigid rotor. When B hr is assumed to be a constant, which is realistic only for symmetrical rotors, equation (8b) simplifies:

& !B hr # 2 +V () # ) *# () =E *# () (9)

Solutions of equation (9) are not given here, but are well known. 70,74,75 For the purpose of calculating densities of states and computing partition functions in the

MultiWell Suite, Eq. (8b) is diagonalized to obtain energy eigenvalues, whether or not B hr is MultiWell Suite, Eq. (8b) is diagonalized to obtain energy eigenvalues, whether or not B hr is

(10) where D is the matrix of the first-order derivative (∂/∂χ) operator of the internal rotation angle,

H = !DB hr D +V=D T B hr D +V

D T is the transpose of D (i.e. D (i,j) = D(j,i)), V(i,i) is the diagonal matrix of the potential energy operator, and B hr (i,i) is the diagonal matrix element of the rotational constant; the indices are for

equally spaced torsion angle grid points. For 2N +1 grid points, the elements of D are:

Di () ,i =0 (11a)

D(i, j) = !1 () { 2 sin #$ ( i!j ) " / 2N ! 1 ( ) %& } for i≠j

i!j

(11b) To construct the symmetric matrix H, one requires both V(χ) and B hr (χ). Several common

representations of V(χ) and B(χ) (or the corresponding moment of inertia function I(χ)) can be understood by MultiWell, as explained elsewhere in this manual. The matrix H is diagonalized in order to obtain a vector of energy eigenvalues, which are convoluted with states from the other degrees of freedom to compute ro-vibrational densities of states or partition functions.

Users must supply the functions V(χ) and B hr (χ) (or moment of inertia function I(χ)). Potential energies V(χ) and molecular geometries can be computed at discrete values of χ by

obtained by using any of the many available quantum chemistry codes, such as Gaussian, 55 C-

56 Four, 60 , and Molpro. . The results for V(χ) can be fitted to a suitable truncated Fourier series, and one may use codes like the I_Eckart program 61 (written for use with MatLab) or LAMM

program given in the MultiWell Suite to compute B hr (χ) or I hr (χ).