GF method

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In chemistry and molecular physics, Wilson's GF method, sometimes referred to as the FG method, is a classical mechanical method to obtain certain internal coordinates for a vibrating semi-rigid molecule, the so-called normal coordinates Q_k (also known as normal modes). Simultaneously, the vibrational energies associated with these normal modes are obtained.

Normal coordinates decouple the classical vibrational motions of the molecule making it possible to compute the vibrational amplitudes of atoms in a molecule. In Wilson's GF method it is assumed that the molecular kinetic energy consists only of vibrations of the atoms, i.e., the overall rotational and translational energies of the molecule are ignored. Normal coordinates are usually defined in a classical mechanical context, but they also appear in the quantum mechanical descriptions of the vibrational motions of molecules and the Coriolis coupling between rotations and vibrations.

It follows from application of the Eckart conditions that the matrix G⁻¹ gives the kinetic energy in terms of arbitrary linear internal coordinates, while F represents the harmonic potential energy in terms of these internal coordinates. Solution of the GF equations gives the matrix that transforms from general internal coordinates to the special set of normal coordinates. The associated eigen energies of the vibrations are obtained along the way.

The method is called after the two matrices F and G that enter the final equations. The name of the most important of its early investigators E. Bright Wilson, Jr. (1908–1992) is often attached to the method.

The GF method

A non-linear molecule consisting of N atoms has 3N−6 internal degrees of freedom, because the positioning of a molecule in three-dimensional space requires three degrees of freedom and the description of its orientation requires another three degree of freedom. These six degrees of freedom must be subtracted from the 3N degrees of freedom of a system of N particles.

The atoms in a molecule are bound by a potential energy surface (PES) (also known as force field) which is a function of 3N−6 coordinates. The internal degrees of freedom q₁, ..., q_3N−6 describing the PES in an optimum way are often non-linear; they are for instance valence coordinates, such as bending and torsion angles and bond stretches. It is possible to write the quantum mechanical kinetic energy operator for such curvilinear coordinates, but it is hard to formulate a general theory applicable to any molecule. This is why Wilson linearized the internal coordinates by assuming small displacements. The linearized version of the internal coordinate q_t is denoted by S_t.

The PES V can be Taylor-expanded around its minimum in terms of the S_t. The third term (the Hessian of V) evaluated in the minimum is a force constant matrix F. In the harmonic approximation—on which the method is based—the Taylor series is ended after this quadratic term. The second term, containing first derivatives, is zero because it is evaluated in the minimum of V. The first term can be included in the zero of energy. Thus,

$2V\approx \sum _{s,t=1}^{3N-6}F_{st}S_{s}\,S_{t}$.

The classical vibrational kinetic energy has the form:

$2T=\sum _{s,t=1}^{3N-6}g_{st}(\mathbf {q} ){\dot {S}}_{s}{\dot {S}}_{t},$

where g_st is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate time derivatives. Evaluation of the metric tensor g(q) in the minimum q₀ of V gives the positive definite and symmetric matrix G ≡ g(q₀)⁻¹.

It is assumed that both F and G can be determined. The (inverse of the) kinetic energy matrix G is usually obtained by analytic methods and the force constant matrix F by fitting to experimental infrared data and/or quantum chemical calculations. Knowing the matrices, one can solve the following two problems simultaneously

$\mathbf {L} ^{\mathrm {T} }\mathbf {F} \mathbf {L} ={\boldsymbol {\Phi }}\quad \mathrm {and} \quad \mathbf {L} ^{\mathrm {T} }\mathbf {G} ^{-1}\mathbf {L} =\mathbf {E} ,$

where $\mathbf {E} \,$ is the identity matrix. This can be done because the two equations are equivalent to the generalized eigenvalue problem

$\mathbf {G} \mathbf {F} \mathbf {L} =\mathbf {L} {\boldsymbol {\Phi }},$

where ${\boldsymbol {\Phi }}=\operatorname {diag} (f_{1},\ldots ,f_{3N-6})$. In numerical mathematics several algorithms are known to solve generalized eigenvalue problems. The solution yields the matrix L and the diagonal matrix Φ that has the vibrational force constants on the diagonal. The matrix L⁻¹ contains the normal coordinates Q_k in its rows:

$Q_{k}=\sum _{t=1}^{3N-6}(\mathbf {L} ^{-1})_{kt}S_{t},\quad k=1,\ldots ,3N-6.\,$

Because of the form of the generalized eigenvalue problem, the method is called the GF method, often with the name of its originator attached to it: Wilson's GF method. It is of interest to point out that by matrix transposition of both sides of the equation and by use of the fact that both G and F are symmetric, (as is a diagonal matrix), one can recast this equation into a very similar one for FG . This is why the method is also referred to as Wilson's FG method.

We introduce the vectors

$\mathbf {s} =\operatorname {col} (S_{1},\ldots ,S_{3N-6})\quad \mathrm {and} \quad \mathbf {Q} =\operatorname {col} (Q_{1},\ldots ,Q_{3N-6}),$

which satisfy the relation

$\mathbf {s} =\mathbf {L} \mathbf {Q} .$

Upon use of the results of the generalized eigenvalue equation the classical (i.e., non-quantum mechanical) energy E = T + V of the molecule becomes in the harmonic approximation,

$2E={\dot {\mathbf {s} }}^{\mathrm {T} }\mathbf {G} ^{-1}{\dot {\mathbf {s} }}+\mathbf {s} ^{\mathrm {T} }\mathbf {F} \mathbf {s}$

$={\dot {\mathbf {Q} }}^{\mathrm {T} }\;\left(\mathbf {L} ^{\mathrm {T} }\mathbf {G} ^{-1}\mathbf {L} \right)\;{\dot {\mathbf {Q} }}+\mathbf {Q} ^{\mathrm {T} }\left(\mathbf {L} ^{\mathrm {T} }\mathbf {F} \mathbf {L} \right)\;\mathbf {Q}$

$={\dot {\mathbf {Q} }}^{\mathrm {T} }{\dot {\mathbf {Q} }}+\mathbf {Q} ^{\mathrm {T} }{\boldsymbol {\Phi }}\mathbf {Q}$

$=\sum _{t=1}^{3N-6}{\big (}{\dot {Q}}_{t}^{2}+f_{t}Q_{t}^{2}{\big )}.$

Solution of the GF equations yields Q_t and f_t. Classically, the time dependence of the harmonic oscillators is given by sine-type functions, so that the time derivatives of the Q_t's are simple cosine-type functions. Hence E is given up to a choice of 3N−6 initial vibrational amplitudes A_t.

The Lagrangian L = T − V is

$L={\frac {1}{2}}\sum _{t=1}^{3N-6}{\big (}{\dot {Q}}_{t}^{2}-f_{t}Q_{t}^{2}{\big )}.$

The corresponding Lagrange equations are identical to a set of harmonic oscillator (Newton) equations

${\ddot {Q}}_{t}+f_{t}\,Q_{t}=0,\quad t=1,2,\ldots ,3N-6.$

In summary, by solving the GF equations the coupled equations have been uncoupled into a set of 3N−6 one-dimensional harmonic oscillator equations. These ordinary second-order differential equations are easily solved yielding Q_t as sine-type functions of time, see the article on harmonic oscillators. Classically, one must further specify amplitudes A_t as initial conditions in order to fix the 3N−6 vibrational energies completely.

The uncoupled harmonic oscillator equations can be solved quantum mechanically as well. The energies are ($\hbar$ is Planck's reduced constant)

$E_{v_{t}}^{(t)}=\hbar {\sqrt {f_{t}}}\;{\big (}v_{t}+{\tfrac {1}{2}}{\big )},\qquad v_{t}=0,1,2,\ldots$

which implies that quantum mechanically 3N−6 vibrational quantum numbers v_t must specified in order to fix the vibrational energy of the molecule completely.

Normal coordinates in terms of Cartesian displacement coordinates

Often the normal coordinates are expressed as linear combinations of Cartesian displacement coordinates. Let R_A be the position vector of nucleus A and R_A⁰ the corresponding equilibrium position. Then $\mathbf {d} _{A}\equiv \mathbf {R} _{A}-\mathbf {R} _{A}^{0}$ is by definition the Cartesian displacement coordinate of nucleus A. Wilson's linearizing of the internal curvilinear coordinates q_t expresses the coordinate S_t in terms of the displacement coordinates

$S_{t}=\sum _{A=1}^{N}\sum _{i=1}^{3}s_{Ai}^{t}\,d_{Ai}=\sum _{A=1}^{N}\mathbf {s} _{A}^{t}\cdot \mathbf {d} _{A},\quad \mathrm {for} \quad t=1,\ldots ,3N-6,$

where s_A^t is known as a Wilson s-vector. If we put the $s_{Ai}^{t}$ into a 3N-6 x 3N matrix B, this equation becomes in matrix language

$\mathbf {s} =\mathbf {B} \mathbf {d} .$

The actual form of the matrix elements of B can be fairly complicated. Especially for a torsion angle, which involves 4 atoms, it requires tedious vector algebra to derive the corresponding values of the $s_{Ai}^{t}$. See for more details on this method, known as the Wilson s-vector method, the book by Wilson et al., or molecular vibration. Now,

$\mathbf {Q} =\mathbf {L} ^{-1}\mathbf {s} =\mathbf {L} ^{-1}\mathbf {B} \mathbf {d} \equiv \mathbf {D} \mathbf {d} .$

In summation language:

$Q_{k}=\sum _{A=1}^{N}\sum _{i=1}^{3}D_{Ai}^{k}\,d_{Ai}\quad \mathrm {for} \quad k=1,\ldots ,3N-6.$

Here D is a 3N-6 x 3N matrix which is given by (i) the linearization of the internal coordinates q (an algebraic process) and (ii) solution of Wilson's GF equations (a numeric process).

Relation with Eckart conditions

From the invariance of the internal coordinates S_t under overall rotation and translation of the molecule, follows the same for the linearized coordinates s^t_A. It can be shown that this implies that the following 6 conditions are satisfied by the internal coordinates,

$\sum _{A=1}^{N}\mathbf {s} _{A}^{t}=0\quad \mathrm {and} \quad \sum _{A=1}^{N}\mathbf {R} _{A}^{0}\times \mathbf {s} _{A}^{t}=0,\quad t=1,\ldots ,3N-6.$

These conditions follow from the Eckart conditions that hold for the displacement vectors,

$\sum _{A=1}^{N}M_{A}\;\mathbf {d} _{A}=0\quad \mathrm {and} \quad \sum _{A=1}^{N}M_{A}\;\mathbf {R} _{A}^{0}\times \mathbf {d} _{A}=0.$

See this article for more details.

References

Cited reference

Further references

• E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955 (Reprinted by Dover 1980).
• D. Papoušek and M. R. Aliev, Molecular Vibrational-Rotational Spectra Elsevier, Amsterdam, 1982.
• S. Califano, Vibrational States, Wiley, London, 1976.