Eckart conditions


Eckart conditions

The Eckart conditions, [ C. Eckart, "Some studies concerning rotating axes and polyatomic molecules", Physical Review,vol. 47, pp. 552-558 (1935).] named after Carl Eckart, sometimes referred to as Sayvetz conditions, [Aaron Sayvetz, "The Kinetic Energy of Polyatomic Molecules", J. Chem. Phys. vol. 7, pp. 383-389 (1939).] simplify the nuclear motion (rovibrational) Schrödinger equation that arises in the second step of the Born-Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external (rotation and translation) motions from the internal (vibration) motions. Although the rotational and vibrational motions of the nuclei in a molecule cannot be fully separated, the Eckart conditions minimize the coupling between these two.

Definition of Eckart conditions

The Eckart conditions can only be formulated for a semi-rigid molecule, which is a molecule with a potential energy surface "V"(R1, R2,..R"N") that has a well-defined minimum for R"A"0 (A=1,ldots, N). These equilibrium coordinates of the nuclei—with masses "M""A"—are expressed with respect to a fixed orthonormal principal axes frame and hence satisfy the relations:sum_{A=1}^N M_A,ig(delta_{ij}|mathbf{R}_A^0|^2 - R^0_{Ai} R^0_{Aj}ig) = lambda^0_i delta_{ij} quadmathrm{and}quadsum_{A=1}^N M_A mathbf{R}_A^0 = mathbf{0}. Here λi0 is a principal inertia moment of the equilibrium molecule.The triplets R"A"0 = ("R""A"10, "R""A"20, "R""A"30) satisfying these conditions, enter the theory as a given set of real constants.Following Biedenharn and Louck [L. C. Biedenharn and J. D. Louck, "Angular Momentum in Quantum Physics", Addison-Wesley, Reading (1981) p. 535.] we introduce an orthonormal body-fixed frame, the "Eckart frame", :vecmathbf{F} = { vec{f}_1, vec{f}_2, vec{f}_3}. If we were tied to the Eckart frame, which—following the molecule—rotates and translates in space, we would observe the molecule in its equilibrium geometry when we would draw the nuclei at the points, :vec{R}_A^0 equiv vecmathbf{F} cdot mathbf{R}_A^0=sum_{i=1}^3 vec{f}_i, R^0_{Ai},quad A=1,ldots,N .Let the elements of R"A" be the coordinates with respect to the Eckart frame of the position vector of nucleus "A" (A=1,ldots, N). Since we take the origin of the Eckart frame in the instantaneous center of mass, the following relation:sum_A M_A mathbf{R}_A = mathbf{0} holds. We define "displacement coordinates" :mathbf{d}_Aequivmathbf{R}_A-mathbf{R}^0_A.Clearly the displacement coordinates satisfy the translational Eckart conditions, :sum_{A=1}^N M_A mathbf{d}_A = 0 .The rotational Eckart conditions for the displacements are::sum_{A=1}^N M_A mathbf{R}^0_A imes mathbf{d}_A = 0, where imes indicates a vector product.These rotational conditions follow from the specific construction of the Eckart frame, see Biedenharn and Louck, "loc. cit.", page 538.

Finally, for a better understanding of the Eckart frame it may be useful to remark that it becomes a principal axes frame in the case that the molecule is a rigid rotor, that is, when all "N" displacement vectors are zero.

eparation of external and internal coordinates

The "N" position vectors vec{R}_A of the nuclei constitute a 3"N" dimensional linear space R3N: the "configuration space". The Eckart conditions give an orthogonal direct sum decomposition of this space:mathbf{R}^{3N} = mathbf{R}_ extrm{ext}oplusmathbf{R}_ extrm{int}.The elements of the 3"N"-6 dimensional subspace Rint are referred to as "internal coordinates", because they are invariant under overall translation and rotation of the molecule and, thus, depend only on the internal (vibrational) motions. The elements of the 6-dimensional subspace Rext are referred to as "external coordinates", because they are associated with the overall translation and rotation of the molecule.

To clarify this nomenclature we define first a basis for Rext. To that end we introduce the following 6 vectors (i=1,2,3)::egin{align}vec{s}^A_{i} &equiv vec{f}_i \vec{s}^A_{i+3} &equiv vec{f}_i imesvec{R}_A^0 .\end{align}An orthogonal, unnormalized, basis for Rext is,:vec{S}_t equiv operatorname{row}(sqrt{M_1};vec{s}^{,1}_{t}, ldots, sqrt{M_N} ;vec{s}^{,N}_{t})quadmathrm{for}quad t=1,ldots, 6.A mass-weighted displacement vector can be written as :vec{D} equiv operatorname{col}(sqrt{M_1};vec{d}^{,1}, ldots, sqrt{M_N};vec{d}^{,N})quadmathrm{with}quadvec{d}^{,A} equiv vec{mathbf{Fcdot mathbf{d}_A .For i=1,2,3, :vec{S}_i cdot vec{D} = sum_{A=1}^N ; M_A vec{s}^{,A}_i cdot vec{d}^{,A}=sum_{A=1}^N M_A d_{Ai} = 0, where the zero follows because of the translational Eckart conditions.For i=4,5,6:, vec{S}_i cdot vec{D} = sum_{A=1}^N ; M_A ig(vec{f}_i imesvec{R}_A^0ig) cdot vec{d}^{,A}=vec{f}_i cdot sum_{A=1}^N M_A vec{R}_A^0 imesvec{d}^A = sum_{A=1}^N M_A ig( mathbf{R}_A^0 imes mathbf{d}_Aig)_i = 0,where the zero follows because of the rotational Eckart conditions. We conclude that the displacement vector vec{D} belongs to the orthogonal complement of Rext, so that it is an internal vector.

We obtain a basis for the internal space by defining 3"N"-6 linearly independent vectors:vec{Q}_r equiv operatorname{row}(frac{1}{sqrt{M_1;vec{q}_r^{,1}, ldots, frac{1}{sqrt{M_N;vec{q}_r^{,N}), quadmathrm{for}quad r=1,ldots, 3N-6.The vectors vec{q}^A_r could be Wilson's s-vectors or could be obtained in the harmonic approximation by diagonalizing the Hessian of "V". We next introduce internal (vibrational) modes,:q_r equiv vec{Q}_r cdot vec{D} = sum_{A=1}^N vec{q}^A_r cdot vec{d}^{,A}quadmathrm{for}quad r=1,ldots, 3N-6.The physical meaning of "q"r depends on the vectors vec{q}^A_r. For instance, "q"r could be a symmetric stretching mode, in which two C—H bonds are simultaneously stretched and contracted.

We already saw that the corresponding external modes are zero because of the Eckart conditions,:s_t equiv vec{S}_t cdot vec{D} = sum_{A=1}^N M_A ;vec{s}^{,A}_t cdot vec{d}^{,A} = 0quadmathrm{for}quad t=1,ldots, 6.

Overall translation and rotation

The vibrational (internal) modes are invariant under translation and infinitesimal rotation of the equilibrium (reference) molecule if and only if the Eckart conditions apply. This will be shown in this subsection.

An overall translation of the reference molecule is given by: vec{R}_{A}^0 mapsto vec{R}_{A}^0 + vec{t} 'for any arbitrary 3-vector vec{t}.An infinitesimal rotation of the molecule is given by: vec{R}_A^0 mapsto vec{R}_A^0 + Deltaphi ; ( vec{n} imes vec{R}_A^0)where Δφ is an infinitesimal angle, Δφ >> (Δφ)², and vec{n} is an arbitrary unit vector. From the orthogonality of vec{Q}_r to the external space follows that the vec{q}^A_r satisfy:sum_{A=1}^N vec{q}^{,A}_r = vec{0} quadmathrm{and}quad sum_{A=1}^N vec{R}^0_A imes vec{q}^A_r = vec{0}. Now, under translation:q_r mapsto sum_Avec{q}^{,A}_r cdot(vec{d}^A - vec{t}) =q_r - vec{t}cdotsum_A vec{q}^{,A}_r = q_r.Clearly, vec{q}^A_r is invariant under translation if and only if :sum_A vec{q}^{,A}_r = 0,because the vector vec{t} is arbitrary. So, the translational Eckart conditions imply the translational invariance of the vectors belonging to internal space and conversely. Under rotation we have,:q_r mapsto sum_Avec{q}^{,A}_r cdot ig(vec{d}^A - Deltaphi ; ( vec{n} imes vec{R}_A^0) ig) =q_r - Deltaphi ; vec{n}cdotsum_A vec{R}^0_A imesvec{q}^{,A}_r = q_r.Rotational invariance follows if and only if :sum_A vec{R}^0_A imesvec{q}^{,A}_r.

The external modes, on the other hand, are "not" invariant and it is not difficult to show that they change under translation as follows::egin{align}s_i &mapsto s_i + M vec{f}_i cdot vec{t} quad mathrm{for}quad i=1,2,3 \s_i &mapsto s_i quad mathrm{for}quad i=4,5,6, \end{align}where "M" is the total mass of the molecule. They change under infinitesimal rotation as follows:egin{align}s_i &mapsto s_i quad mathrm{for}quad i=1,2,3 \s_i &mapsto s_i + Delta phi vec{f}_i cdot mathbf{I}^0cdot vec{n} quad mathrm{for}quad i=4,5,6, \end{align}where I0 is the inertia tensor of the equilibrium molecule. This behavior showsthat the first three external modes describe the overall translation of the molecule, whilethe modes 4, 5, and, 6 describe the overall rotation.

Vibrational energy

The vibrational energy of the molecule can be written in terms of coordinates with respect to the Eckart frame as:2T_mathrm{vib} = sum_{A=1}^N M_A dot{mathbf{R_Acdot dot{mathbf{R_A= sum_{A=1}^N M_A dot{mathbf{d_Acdot dot{mathbf{d_A.Because the Eckart frame is non-inertial, the total kinetic energy comprises also centrifugal and Coriolis energies. These stay out of the present discussion. The vibrational energy is written in terms of the displacement coordinates, which are linearly dependent because they are contaminated by the 6 external modes, which are zero, i.e., the d"A"'s satisfy 6 linear relations. It is possible to write the vibrational energy solely in terms of the internal modes "q"r ("r" =1, ..., 3"N"-6) as we will now show. We write the different modes in terms of the displacements:egin{align}q_r = sum_{Aj} d_{Aj}& ig( q^A_{rj} ig) \s_i = sum_{Aj} d_{Aj}& ig( M_A delta_{ij} ig) =0 \s_{i+3} = sum_{Aj} d_{Aj}& ig( M_A sum_k epsilon_{ikj} R^0_{Ak} ig)=0 \end{align}The parenthesized expressions define a matrix B relating the internal and external modes to the displacements. The matrix B may be partitioned in an internal (3"N"-6 x 3"N") and an external (6 x 3"N") part,:mathbf{v}equiv egin{pmatrix}q_1 \vdots \vdots \q_{3N-6} \0 \vdots \0\end{pmatrix}= egin{pmatrix}mathbf{B}^mathrm{int} \cdots \mathbf{B}^mathrm{ext} \end{pmatrix}mathbf{d} equiv mathbf{B} mathbf{d}.We define the matrix M by:mathbf{M} equiv operatorname{diag}(mathbf{M}_1, mathbf{M}_2, ldots,mathbf{M}_N)quad extrm{and}quadmathbf{M}_Aequiv operatorname{diag}(M_A, M_A, M_A)and from the relations given in the previous sections follow the matrix relations:mathbf{B}^mathrm{ext} mathbf{M}^{-1} (mathbf{B}^mathrm{ext})^mathrm{T}= operatorname{diag}(N_1,ldots, N_6) equivmathbf{N},and:mathbf{B}^mathrm{int} mathbf{M}^{-1} (mathbf{B}^mathrm{ext})^mathrm{T}= mathbf{0}.We define:mathbf{G} equivmathbf{B}^mathrm{int} mathbf{M}^{-1} (mathbf{B}^mathrm{int})^mathrm{T}.By using the rules for block matrix multiplication we can show that:(mathbf{B}^mathrm{T})^{-1} mathbf{M} mathbf{B}^{-1}= egin{pmatrix}mathbf{G}^{-1} && mathbf{0} \mathbf{0} && mathbf{N}^{-1}end{pmatrix},where G-1 is of dimension (3"N"-6 x 3"N"-6) and N-1 is (6 x 6).The kinetic energy becomes:2T_mathrm{vib} = dot{mathbf{d^mathrm{T} mathbf{M} dot{mathbf{d= dot{mathbf{v^mathrm{T}; (mathbf{B}^mathrm{T})^{-1} mathbf{M} mathbf{B}^{-1}; dot{mathbf{v = sum_{r, r'=1}^{3N-6} (G^{-1})_{r r'} dot{q}_r dot{q}_{r'}where we used that the last 6 components of v are zero. This form ofthe kinetic energy of vibration enters Wilson's GF method. It is of some interest to point out that the potential energy in the harmonic approximation can be written as follows:2V_mathrm{harm} = mathbf{d}^mathrm{T} mathbf{H} mathbf{d}= mathbf{v}^mathrm{T} (mathbf{B}^mathrm{T})^{-1} mathbf{H} mathbf{B}^{-1} mathbf{v} = sum_{r, r'=1}^{3N-6} F_{r r'} q_r q_{r'},where H is the Hessian of the potential in the minimum and F, defined by this equation, is the F matrix of the GF method.

Relation to the harmonic approximation

In the harmonic approximation to the nuclear vibrational problem, expressed in displacement coordinates, one must solve the generalized eigenvalue problem: mathbf{H}mathbf{C} = mathbf{M} mathbf{C} oldsymbol{Phi},where H is a 3"N" x 3"N" symmetric matrix of second derivatives of the potential V(mathbf{R}_1, mathbf{R}_2,ldots, mathbf{R}_N). H is the Hessian matrix of "V" in the equilibrium mathbf{R}_1^0,ldots, mathbf{R}_N^0. The diagonal matrix M contains the masses on the diagonal. The diagonal matrix oldsymbol{Phi} contains the eigenvalues, whilethe columns of C contain the eigenvectors.

It can be shown that the invariance of "V" under simultaneous translation over t of all nuclei implies that vectors T = (t, ... , t) are in the kernel of H.From the invariance of "V" under an infinitesimal rotation of all nuclei around s it can been shown that also the vectors S = (s x R10, ..., s x RN0) are in the kernel of H : :mathbf{H}egin{pmatrix} mathbf{t} \ vdots\ mathbf{t} end{pmatrix} =egin{pmatrix} mathbf{0} \ vdots\ mathbf{0} end{pmatrix}quadmathrm{and}quadmathbf{H}egin{pmatrix} mathbf{s} imes mathbf{R}_1^0 \ vdots\ mathbf{s} imes mathbf{R}_N^0 end{pmatrix} =egin{pmatrix} mathbf{0} \ vdots\ mathbf{0} end{pmatrix} Thus, 6 columns of C corresponding to eigenvalue zero, are determined algebraically. (If the generalized eigenvalue problem is solved numerically, one will find in general 6 linearly independent linear combinations of S and T). The eigenspace corresponding to eigenvalue zero is at least of dimension 6 (often it is exactly of dimension 6, since the other eigenvalues, which are force constants, are never zero for molecules in their ground state). Thus, T and S correspond to the overall (external) motions: translation and rotation, respectively. They are "zero-energy modes" because space is homogeneous (force-free) and isotropic (torque-free).

By the definition in this article the non-zero frequency modes are internal modes, since they are within the orthogonal complement of Rext. The generalized orthogonalities: mathbf{C}^mathrm{T} mathbf{M} mathbf{C} = mathbf{I} applied to the "internal" (non-zero eigenvalue) and "external" (zero-eigenvalue) columns of Care equivalent to the Eckart conditions.

References

The classic work is:
* E. Bright Wilson, Jr., J. C. Decius and Paul C. Cross, "Molecular Vibrations", Mc-Graw-Hill (1955). Reprinted by Dover (1980).

More advanced book are:
* D. Papoušek and M. R. Aliev, "Molecular Vibrational-Rotational Spectra", Elsevier (1982).
* S. Califano, "Vibrational States", Wiley, New York-London (1976). ISBN 0 471 12996 8

External links

* For a free code (developed by Z. Szabo and R. Scipioni) which calculates the vibrational frequencies starting from the Force Constant matrix in mass weighted coordinates see: [http://www.evtsz.bme.hu/web/staff/szabo/web_molecular_vibration/molec_vib_code.html Molecular Vibration ]


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