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GF method
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<h2 id="the-gf-method">The GF method</h2> <p>A non-linear molecule consisting of <i>N</i> atoms has 3<i>N</i> − 6 internal <a href="/facts/Degrees_of_freedom_(physics_and_chemistry)/Rrjl0lnq">degrees of freedom</a>, because positioning a molecule in three-dimensional space requires three degrees of freedom, and the description of its orientation in space requires another three degree of freedom. These degrees of freedom must be subtracted from the 3<i>N</i> degrees of freedom of a system of <i>N</i> particles. </p><p>The interaction among atoms in a molecule is described by a <a href="/facts/Potential_energy_surface/YDUNcSM8">potential energy surface</a> (PES), which is a function of 3<i>N</i> − 6 coordinates. The internal degrees of freedom <i>s</i>1, ..., <i>s</i>3<i>N</i>−6 describing the PES in an optimal way are often non-linear; they are for instance <i>valence coordinates</i>, such as bending and torsion angles and bond stretches. It is possible to write the quantum mechanical kinetic energy operator for such <a href="/facts/Curvilinear_coordinates/RQD7Cajd">curvilinear coordinates</a>, 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.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> The linearized version of the internal coordinate <i>s</i>t is denoted by <i>S</i>t. </p><p>The PES <i>V</i> can be Taylor expanded around its minimum in terms of the <i>S</i>t. The third term (the <a href="/facts/Hessian_matrix/RguMNr3m">Hessian</a> of <i>V</i>) evaluated in the minimum is a force derivative matrix F. In the harmonic approximation the <a href="/facts/Taylor_series/M0aTuftH">Taylor series</a> is ended after this term. The second term, containing first derivatives, is zero because it is evaluated in the minimum of <i>V</i>. The first term can be included in the zero of energy. Thus, </p> 2 V ≈ ∑ s , t = 1 3 N − 6 F s t S s S t . {\displaystyle 2V\approx \sum _{s,t=1}^{3N-6}F_{st}S_{s}\,S_{t}.} <p>The classical vibrational kinetic energy has the form: </p> 2 T = ∑ s , t = 1 3 N − 6 g s t ( s ) S ˙ s S ˙ t , {\displaystyle 2T=\sum _{s,t=1}^{3N-6}g_{st}(\mathbf {s} ){\dot {S}}_{s}{\dot {S}}_{t},} <p>where <i>g</i><i>st</i> is an element of the metric tensor of the internal (curvilinear) coordinates. The dots indicate <a href="/facts/Time_derivative/AWV28T13">time derivatives</a>. Mixed terms S s S ˙ t {\displaystyle S_{s}\,{\dot {S}}_{t}} generally present in curvilinear coordinates are not present here, because only linear coordinate transformations are used. Evaluation of the metric tensor g in the minimum s0 of <i>V</i> gives the positive definite and symmetric matrix G = g(s0)−1. One can solve the two matrix problems </p> L T F L = Φ a n d L T G − 1 L = E , {\displaystyle \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} ,} <p>simultaneously, since they are equivalent to the <a href="/facts/Generalized_eigenvalue_problem/a2nfF7hJ">generalized eigenvalue problem</a> </p> G F L = L Φ , {\displaystyle \mathbf {G} \mathbf {F} \mathbf {L} =\mathbf {L} {\boldsymbol {\Phi }},} <p>where Φ = diag ( f 1 , … , f 3 N − 6 ) {\displaystyle {\boldsymbol {\Phi }}=\operatorname {diag} (f_{1},\ldots ,f_{3N-6})} where <i>fi</i> is equal to 4 π 2 ν i 2 {\displaystyle 4{\pi }^{2}{\nu }_{i}^{2}} ( ν i {\displaystyle {\nu }_{i}} is the frequency of normal mode <i>i</i>); E {\displaystyle \mathbf {E} \,} is the unit matrix. The matrix L−1 contains the <i>normal coordinates</i> <i>Q</i>k in its rows: </p> Q k = ∑ t = 1 3 N − 6 ( L − 1 ) k t S t , k = 1 , … , 3 N − 6. {\displaystyle Q_{k}=\sum _{t=1}^{3N-6}(\mathbf {L} ^{-1})_{kt}S_{t},\quad k=1,\ldots ,3N-6.\,} <p>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. By matrix transposition in both sides of the equation and using the fact that both G and F are symmetric matrices, as are diagonal matrices, 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. </p><p>We introduce the vectors </p> s = col ( S 1 , … , S 3 N − 6 ) a n d Q = col ( Q 1 , … , Q 3 N − 6 ) , {\displaystyle \mathbf {s} =\operatorname {col} (S_{1},\ldots ,S_{3N-6})\quad \mathrm {and} \quad \mathbf {Q} =\operatorname {col} (Q_{1},\ldots ,Q_{3N-6}),} <p>which satisfy the relation </p> s = L Q . {\displaystyle \mathbf {s} =\mathbf {L} \mathbf {Q} .} <p>Upon use of the results of the generalized eigenvalue equation, the energy <i>E</i> = <i>T </i> + <i>V</i> (in the harmonic approximation) of the molecule becomes: </p> 2 E = s ˙ T G − 1 s ˙ + s T F s {\displaystyle 2E={\dot {\mathbf {s} }}^{\mathrm {T} }\mathbf {G} ^{-1}{\dot {\mathbf {s} }}+\mathbf {s} ^{\mathrm {T} }\mathbf {F} \mathbf {s} } = Q ˙ T ( L T G − 1 L ) Q ˙ + Q T ( L T F L ) Q {\displaystyle ={\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} } = Q ˙ T Q ˙ + Q T Φ Q = ∑ t = 1 3 N − 6 ( Q ˙ t 2 + f t Q t 2 ) . {\displaystyle ={\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 )}.} <p>The Lagrangian <i>L</i> = <i>T</i> − <i>V</i> is </p> L = 1 2 ∑ t = 1 3 N − 6 ( Q ˙ t 2 − f t Q t 2 ) . {\displaystyle L={\frac {1}{2}}\sum _{t=1}^{3N-6}{\big (}{\dot {Q}}_{t}^{2}-f_{t}Q_{t}^{2}{\big )}.} <p>The corresponding <a href="/facts/Lagrange_equations/sTxyqWz8">Lagrange equations</a> are identical to the Newton equations </p> Q ¨ t + f t Q t = 0 {\displaystyle {\ddot {Q}}_{t}+f_{t}\,Q_{t}=0} <p>for a set of uncoupled harmonic oscillators. These ordinary second-order differential equations are easily solved, yielding <i>Q</i><i>t</i> as a function of time; see the article on <a href="/facts/Harmonic_oscillator/7IQNToGt">harmonic oscillators</a>. </p> <h2 id="normal-coordinates-in-terms-of-cartesian-displacement-coordinates">Normal coordinates in terms of Cartesian displacement coordinates</h2> <p>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 x A ≡ R A − R A 0 {\displaystyle \mathbf {x} _{A}\equiv \mathbf {R} _{A}-\mathbf {R} _{A}^{0}} is by definition the <i>Cartesian displacement coordinate</i> of nucleus A. Wilson's linearizing of the internal curvilinear coordinates <i>q</i><i>t</i> expresses the coordinate <i>S</i><i>t</i> in terms of the displacement coordinates </p> S t = ∑ A = 1 N ∑ i = 1 3 s A i t x A i = ∑ A = 1 N s A t ⋅ x A , f o r t = 1 , … , 3 N − 6 , {\displaystyle S_{t}=\sum _{A=1}^{N}\sum _{i=1}^{3}s_{Ai}^{t}\,x_{Ai}=\sum _{A=1}^{N}\mathbf {s} _{A}^{t}\cdot \mathbf {x} _{A},\quad \mathrm {for} \quad t=1,\ldots ,3N-6,} <p>where sAt is known as a <i>Wilson s-vector</i>. If we put the s A i t {\displaystyle s_{Ai}^{t}} into a (3<i>N</i> − 6) × 3<i>N</i> matrix B, this equation becomes in matrix language </p> s = B x . {\displaystyle \mathbf {s} =\mathbf {B} \mathbf {x} .} <p>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 A i t {\displaystyle s_{Ai}^{t}} . See for more details on this method, known as the <i>Wilson s-vector method</i>, the book by Wilson <i>et al.</i>, or <a href="/facts/Molecular_vibration/Y0ovRukI">molecular vibration</a>. Now, </p> s = L Q = L l t r q = B M − 1 / 2 q ≡ D q , {\displaystyle \mathbf {s} =\mathbf {L} \mathbf {Q} =\mathbf {L} \mathbf {l} ^{tr}\mathbf {q} =\mathbf {B} \mathbf {M} ^{-1/2}\mathbf {q} \equiv \mathbf {D} \mathbf {q} ,} <p>which can be inverted and put in summation language: </p> Q k = ∑ A = 1 N ∑ i = 1 3 D A i k d A i f o r k = 1 , … , 3 N − 6. {\displaystyle Q_{k}=\sum _{A=1}^{N}\sum _{i=1}^{3}D_{Ai}^{k}\,d_{Ai}\quad \mathrm {for} \quad k=1,\ldots ,3N-6.} <p>Here D is a (3<i>N</i> − 6) × 3<i>N</i> matrix, which is given by (i) the linearization of the internal coordinates s (an algebraic process) and (ii) solution of Wilson's GF equations (a numeric process). </p> <h2 id="matrices-involved-in-the-analysis">Matrices involved in the analysis</h2> <p>There are several related coordinate systems commonly used in the GF matrix analysis.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> These quantities are related by a variety of matrices. For clarity, we provide the coordinate systems and their interrelations here. </p><p>The relevant coordinates are: </p> <ul><li> x : {\displaystyle \mathbf {x} :} Cartesian coordinates for each atom</li> <li> s : {\displaystyle \mathbf {s} :} Internal coordinates for each atom</li> <li> q : {\displaystyle \mathbf {q} :} Mass-weighted Cartesian coordinates</li> <li> Q : {\displaystyle \mathbf {Q} :} Normal coordinates</li></ul> <p>These different coordinate systems are related to one another by: </p> <ul><li> s = B x {\displaystyle \mathbf {s} =\mathbf {B} \mathbf {x} } , i.e. the matrix B {\displaystyle \mathbf {B} } transforms the Cartesian coordinates to (linearized) internal coordinates.</li> <li> x = M − 1 / 2 q , {\displaystyle \mathbf {x} =\mathbf {M} ^{-1/2}\mathbf {q} ,} i.e. the mass matrix M 1 / 2 {\displaystyle \mathbf {M} ^{1/2}} transforms Cartesian coordinates to mass-weighted Cartesian coordinates.</li> <li> q = l Q , {\displaystyle \mathbf {q} =\mathbf {l} \mathbf {Q} ,} i.e. the matrix l {\displaystyle \mathbf {l} } transforms the normal coordinates to mass-weighted Cartesian coordinates.</li> <li> s = L Q , {\displaystyle \mathbf {s} =\mathbf {L} \mathbf {Q} ,} i.e. the matrix L {\displaystyle \mathbf {L} } transforms the normal coordinates to internal coordinates.</li></ul> <p>Note the useful relationship: </p><p> L = B M − 1 / 2 l . {\displaystyle \mathbf {L} =\mathbf {B} \mathbf {M} ^{-1/2}\mathbf {l} .} </p><p>These matrices allow one to construct the G matrix quite simply as </p><p> G = B M − 1 B T . {\displaystyle \mathbf {G} =\mathbf {B} \mathbf {M} ^{-1}\mathbf {B} ^{\rm {T}}.} </p> <h2 id="relation-to-eckart-conditions">Relation to Eckart conditions</h2> <p class="note">Main article: <a href="/facts/Eckart_conditions/r2Xhy47L">Eckart conditions</a></p> <p>From the invariance of the internal coordinates <i>S</i><i>t</i> under overall rotation and translation of the molecule, follows the same for the linearized coordinates s<i>t</i>A. It can be shown that this implies that the following 6 conditions are satisfied by the internal coordinates, </p> ∑ A = 1 N s A t = 0 a n d ∑ A = 1 N R A 0 × s A t = 0 , t = 1 , … , 3 N − 6. {\displaystyle \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.} <p>These conditions follow from the Eckart conditions that hold for the displacement vectors, </p> ∑ A = 1 N M A d A = 0 a n d ∑ A = 1 N M A R A 0 × d A = 0. {\displaystyle \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.} <h2 id="further-references">Further references</h2> <ul><li>Califano, S. (1976). <i>Vibrational States</i>. New York-London: Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-12996-8.</li> <li>Papoušek, D.; Aliev, M. R. (1982). <i>Molecular Vibrational-Rotational Spectra</i>. <a href="/facts/Elsevier/jkPSAEmR">Elsevier</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-444-99737-7.</li> <li>Wilson, E. B.; Decius, J. C.; Cross, P. C. (1995) [1955]. <a href="https://archive.org/details/molecularvibrati00wils"><i>Molecular Vibrations</i></a>. New York: Dover. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-63941-X.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Wilson, E. B. Jr. (1941). "Some Mathematical Methods for the Study of Molecular Vibrations". J. Chem. Phys. 9 (1): 76–84. Bibcode:1941JChPh...9...76W. doi:10.1063/1.1750829. <a href="/wiki/Journal_of_Chemical_Physics" target="_blank">/wiki/Journal_of_Chemical_Physics</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Califano, S. (1976). Vibrational states. London: Wiley. ISBN 0-471-12996-8. OCLC 1529286. <a href="0-471-12996-8" target="_blank">0-471-12996-8</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>