GF method: Difference between revisions
imported>Paul Wormer m (links to harmonic oscillator (quantum)) |
imported>Paul Wormer (→The GF method: Clarified the method after some valid WP criticism) |
||
Line 10: | Line 10: | ||
== The GF method == | == The GF method == | ||
A non-linear molecule consisting of ''N'' atoms has 3''N'' | A non-linear molecule consisting of ''N'' atoms has 3''N''−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 3''N'' degrees of freedom of a system of ''N'' particles. | ||
positioning a molecule in three-dimensional space requires three degrees of freedom and the description of its orientation | |||
The atoms in a molecule are bound by a [[potential energy surface]] (PES) ( | The atoms in a molecule are bound by a [[potential energy surface]] (PES) (also known as [[force field (chemistry)|force field]]) which is a function of 3''N''−6 coordinates. | ||
The internal degrees of freedom ''q''<sub>1</sub>, ..., ''q''<sub>3''N'' | The internal degrees of freedom ''q''<sub>1</sub>, ..., ''q''<sub>3''N''−6</sub> 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<ref name="Wilson41"/> linearized the internal coordinates by assuming small displacements. The linearized version of the internal coordinate ''q''<sub>''t''</sub> is denoted by ''S''<sub>''t''</sub>. | ||
The PES ''V'' can be [[Taylor series|Taylor-expanded]] around its minimum in terms of the ''S''<sub>''t''</sub>. The third term (the [[Hessian matrix|Hessian]] of ''V'') evaluated in the minimum is a force | The PES ''V'' can be [[Taylor series|Taylor-expanded]] around its minimum in terms of the ''S''<sub>''t''</sub>. The third term (the [[Hessian matrix|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, | Thus, | ||
:<math> 2V \approx \sum_{s,t=1}^{3N-6} F_{st} S_s\, S_t </math>. | :<math> 2V \approx \sum_{s,t=1}^{3N-6} F_{st} S_s\, S_t </math>. | ||
Line 22: | Line 21: | ||
The classical vibrational kinetic energy has the form: | The classical vibrational kinetic energy has the form: | ||
:<math> 2T = \sum_{s,t=1}^{3N-6} g_{st}(\mathbf{q}) \dot{S}_s\dot{S}_t ,</math> | :<math> 2T = \sum_{s,t=1}^{3N-6} g_{st}(\mathbf{q}) \dot{S}_s\dot{S}_t ,</math> | ||
where ''g''<sub>''st''</sub> is an element of the [[metric tensor]] of the internal (curvilinear) coordinates. The dots indicate [[time derivative]]s. Evaluation of the metric tensor '''g''' in the minimum '''q'''< | where ''g''<sub>''st''</sub> is an element of the [[metric tensor]] of the internal (curvilinear) coordinates. The dots indicate [[time derivative]]s. Evaluation of the metric tensor '''g'''('''q''') in the minimum '''q'''<sub>0</sub> of ''V'' gives | ||
the [[positive define matrix|positive definite]] and [[symmetric matrix]] '''G''' | the [[positive define matrix|positive definite]] and [[symmetric matrix]] '''G''' ≡ '''g'''('''q'''<sub>0</sub>)<sup>−1</sup>. | ||
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 | |||
:<math> \mathbf{L}^\mathrm{T} \mathbf{F} \mathbf{L} =\boldsymbol{\Phi} | :<math> \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}, | \quad \mathrm{and}\quad \mathbf{L}^\mathrm{T} \mathbf{G}^{-1} \mathbf{L} = \mathbf{E}, | ||
</math> | </math> | ||
where <math>\mathbf{E}\,</math> is the [[identity matrix]]. | |||
This can be done because the two equations are equivalent to the [[generalized eigenvalue problem]] | |||
:<math> | :<math> | ||
\mathbf{G} \mathbf{F} \mathbf{L} = \mathbf{L} \boldsymbol{\Phi}, | \mathbf{G} \mathbf{F} \mathbf{L} = \mathbf{L} \boldsymbol{\Phi}, | ||
</math> | </math> | ||
where <math>\boldsymbol{\Phi}=\operatorname{diag}(f_1,\ldots, f_{3N-6}) </math> and | where <math>\boldsymbol{\Phi}=\operatorname{diag}(f_1,\ldots, f_{3N-6}) </math>. | ||
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 constant]]s on the diagonal. | |||
The matrix '''L'''<sup>−1</sup> contains the ''normal coordinates'' ''Q''<sub>k</sub> in its rows: | |||
:<math> Q_k = \sum_{t=1}^{3N-6} (\mathbf{L}^{-1})_{kt} S_t , \quad k=1,\ldots, 3N-6. \,</math> | :<math> Q_k = \sum_{t=1}^{3N-6} (\mathbf{L}^{-1})_{kt} S_t , \quad k=1,\ldots, 3N-6. \,</math> | ||
Because of the form of the generalized eigenvalue problem, the method is called the GF method, | 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'''. | 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 | 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'''. | ||
is also referred to as '''Wilson's FG method''' | |||
We introduce the vectors | We introduce the vectors | ||
Line 48: | Line 50: | ||
\mathbf{s} = \mathbf{L} \mathbf{Q}. | \mathbf{s} = \mathbf{L} \mathbf{Q}. | ||
</math> | </math> | ||
Upon use of the results of the generalized eigenvalue equation | 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, | ||
the energy ''E'' = ''T '' + ''V'' | |||
:<math> | :<math> | ||
2E = \dot{\mathbf{s}}^\mathrm{T} \mathbf{G}^{-1}\dot{\mathbf{s}}+ | 2E = \dot{\mathbf{s}}^\mathrm{T} \mathbf{G}^{-1}\dot{\mathbf{s}}+ | ||
Line 62: | Line 63: | ||
::<math> | ::<math> | ||
= \dot{\mathbf{Q}}^\mathrm{T}\dot{\mathbf{Q}} + \mathbf{Q}^\mathrm{T}\boldsymbol{\Phi}\mathbf{Q} | = \dot{\mathbf{Q}}^\mathrm{T}\dot{\mathbf{Q}} + \mathbf{Q}^\mathrm{T}\boldsymbol{\Phi}\mathbf{Q} | ||
</math> | |||
::<math> | |||
= \sum_{t=1}^{3N-6} \big( \dot{Q}_t^2 + f_t Q_t^2 \big). | = \sum_{t=1}^{3N-6} \big( \dot{Q}_t^2 + f_t Q_t^2 \big). | ||
</math> | </math> | ||
Solution of the GF equations yields ''Q''<sub>''t''</sub> and ''f''<sub>''t''</sub>. | Solution of the GF equations yields ''Q''<sub>''t''</sub> and ''f''<sub>''t''</sub>. Classically, the time dependence of the [[harmonic oscillator (classical)|harmonic oscillator]]s is given by sine-type functions, so that the time derivatives of the ''Q''<sub>''t''</sub>'s are simple cosine-type functions. Hence ''E'' is given up to a choice of 3''N''−6 initial vibrational amplitudes ''A''<sub>''t''</sub>. | ||
The Lagrangian ''L'' = ''T'' | The Lagrangian ''L'' = ''T'' − ''V'' is | ||
:<math> | :<math> | ||
L = \frac{1}{2} \sum_{t=1}^{3N-6} \big( \dot{Q}_t^2 - f_t Q_t^2 \big). | L = \frac{1}{2} \sum_{t=1}^{3N-6} \big( \dot{Q}_t^2 - f_t Q_t^2 \big). | ||
</math> | </math> | ||
The corresponding [[Lagrange equations]] are identical to the | The corresponding [[Lagrange equations]] are identical to a set of [[harmonic oscillator (classical)|harmonic oscillator]] (Newton) equations | ||
:<math> | |||
\ddot{Q}_t + f_t \,Q_t = 0, \quad t=1,2,\ldots, 3N-6. | |||
</math> | |||
In summary, by solving the GF equations the coupled equations have been uncoupled into a set of 3''N''−6 one-dimensional harmonic oscillator equations. These ordinary second-order differential equations are easily solved yielding ''Q''<sub>''t''</sub> as sine-type functions of time, see the article on [[harmonic oscillator (classical)|harmonic oscillator]]s. Classically, one must further specify amplitudes ''A''<sub>''t''</sub> as initial conditions in order to fix the 3''N''−6 vibrational energies completely. | |||
The uncoupled harmonic oscillator equations can be solved quantum mechanically as well. The energies are (<math>\hbar</math> is [[Planck's constant|Planck's reduced constant]]) | |||
:<math> | :<math> | ||
\ | E^{(t)}_{v_t} = \hbar \sqrt{f_t}\;\big(v_t+\tfrac{1}{2}\big), \qquad v_t=0,1,2, \ldots | ||
</math> | </math> | ||
which implies that quantum mechanically 3''N''−6 vibrational quantum numbers ''v''<sub>''t''</sub> must specified in order to fix the vibrational energy of the molecule completely. | |||
==Normal coordinates in terms of Cartesian displacement coordinates == | ==Normal coordinates in terms of Cartesian displacement coordinates == |
Revision as of 09:52, 27 January 2009
![](http://s9.addthis.com/button1-share.gif)
- The content on this page originated on Wikipedia and is yet to be significantly improved. Contributors are invited to replace and add material to make this an original article.
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 Qk (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[1] 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, ..., q3N−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[1] linearized the internal coordinates by assuming small displacements. The linearized version of the internal coordinate qt is denoted by St.
The PES V can be Taylor-expanded around its minimum in terms of the St. 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,
- .
The classical vibrational kinetic energy has the form:
where gst 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
where is the identity matrix. This can be done because the two equations are equivalent to the generalized eigenvalue problem
where . 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 Qk in its rows:
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
which satisfy the relation
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,
Solution of the GF equations yields Qt and ft. Classically, the time dependence of the harmonic oscillators is given by sine-type functions, so that the time derivatives of the Qt's are simple cosine-type functions. Hence E is given up to a choice of 3N−6 initial vibrational amplitudes At.
The Lagrangian L = T − V is
The corresponding Lagrange equations are identical to a set of harmonic oscillator (Newton) equations
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 Qt as sine-type functions of time, see the article on harmonic oscillators. Classically, one must further specify amplitudes At 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 ( is Planck's reduced constant)
which implies that quantum mechanically 3N−6 vibrational quantum numbers vt 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 RA be the position vector of nucleus A and RA0 the corresponding equilibrium position. Then is by definition the Cartesian displacement coordinate of nucleus A. Wilson's linearizing of the internal curvilinear coordinates qt expresses the coordinate St in terms of the displacement coordinates
where sAt is known as a Wilson s-vector. If we put the into a 3N-6 x 3N matrix B, this equation becomes in matrix language
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 . See for more details on this method, known as the Wilson s-vector method, the book by Wilson et al., or molecular vibration. Now,
In summation language:
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 St under overall rotation and translation of the molecule, follows the same for the linearized coordinates stA. It can be shown that this implies that the following 6 conditions are satisfied by the internal coordinates,
These conditions follow from the Eckart conditions that hold for the displacement vectors,
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.