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Molecule-fixed axes in many-body Hamiltonians

Mirjana Mladenović

Published Jan 8, 2020• 0 citations
OpenAlex

Abstract

The separation of the dynamics of the quantum many-body systems into the overall rotation and internal motions leads to their quantization in rotating frames. Whereas the rotating body frame is unique for a rigid body, there is a great degree of freedom for choosing a body frame for a general rotating and vibrating N-particle system. This, in turn, implies the need to study many-body systems for a class of body frames.The concept of vibrational coordinates and of rotational coordinates is particularly clear for the case of a triatomic molecule: three particles form a single plane, in which the molecule vibrates, and at the same time the molecular plane rotates in the space-fixed frame of reference. The overall molecular rotation is made explicit by means of an orthogonal transformation, which brings the space-fixed into the body-fixed axes. The direction perpendicular to the molecular plane uniquely determines one of the three axes of the frame. Rotation about this direction will change the position of the axes in the molecular plane, in fact the axis system. It is already intuitively clear that this axis transformation will not affect the vibrational portion of the kinetic energy operator. The other parts of the kinetic energy, the rotational part and the rotation-vibration coupling part, will then need to adjust themselves to the new definition of the body frame. In this fashion, we may address the problem of the interaction of rotations and vibrations, which is one of the most important problems in molecular studies.The variation of the rovibrational kinetic energy operator following a change of orientation of the axis system is investigated in the present work from the physical and mathematical points of view. Various types of axis embedding are analyzed in a systematic fashion for triatomic molecules of arbitrary structure, including the bisector, principal axis, and Eckart frames. We also consider the bisector axis embedding for tetratomic molecules and the local axis embedding for pentatomic molecules. Different axis conventions lead sometimes to a complex entanglement of rotational and vibrational motions, which may pose serious obstacles to numerically exact approaches.

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Abstract The years ago Kramers and pauli quantized the diatomic molecular model which bears their names, obtaining the formula which has placed so large a part in hand spectroscopy. It is sometimes pointed out in modern text-books that these results, founded on the old mechanics, do not offer a very far-reaching analogy to those which the wave mechanics give for the real diatomic molecule. I think the reason will be seen on closely examining the classical calculation, either in the original papers, or in Born's "Atom-mechanik," Without adequate cause, all types of motion save one were rejected in the quantization; all that remained was the simple type of motion which, apart from the gyroscope's axial spin consists merely of a steady precession of the body as a whole about the invariable axis. It will here be shown that in reality we can have Kramers and Pauli models performing a very different type of motion, while completely satisfying the old-quantum equations (which were determined in the original papers). In these motions the gyroscope presesses not only about the invariable axis but also about the nuclear axis. Kemble proposed that in a modified model, having an elastically mounted gyroscope, notions of this more complex type should be possible as quantized States, though his actual treatment is confined to motions of the simpler type. As stated above, we shall see that the original model can have the more complex type of motion, in strictly quantized states. Kemble also proposed an energy formula for the corresponding motions of the real molecule; it is with this and with the well-established molecular formulæ of the wave mechanics that results obtained from the model are to be compared. For specifying orders of magnitude it is convenient to think what happens when the mass of the nuclei is made very great, their angular momentum retaining the same order of magnitude as that of the gyroscope. Then the existence of quantized motions other than those prescribed by Kramers and Pauli is not surprising; for by analogy with the vectorial conception of the real molecular (described, notably, by Hund and Weizel), we should expext, in these circumstances, to find models in whose quantized states the nuclei, if of great mass, will rotate about the invariable axis very s;lowly in comparison with the gyroscope's precession about the nuclear axis. For the real molecule, wave mechanics equations such as Kronig's (22) will hold with an accuracy increasing with the nuclear mass, in the sense that though the main rotational terms will became smaller, the neglected perturbations (if so they may be called) with diminish far more rapidly.

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Quantum mechanics for internal motions of the three-body system is set up on the basis of the complex vector bundle theory. The three-body system is called a triatomic molecule in the Born–Oppenheimer approximation. The internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator. This bundle is equipped with a linear connection, which is a natural consequence of a geometric interpretation of the so-called Eckart condition. The coupling of the internal motion with the rotation is understood naturally in terms of this connection. The internal Hamiltonian operator is obtained which includes the internal motion–rotation coupling and a centrifugal potential. The complex vector bundle for the triatomic molecule proves to be a trivial bundle, though the geometric setting for the internal motion is independent of whether the bundle is trivial or not.

Journal of Mathematical Physics•Jun 1, 1987•78 citations
The separation of the dynamics of the quantum many-body systems into the overall rotation and internal motions leads to their quantization in rotating frames. Whereas the rotating body frame is unique for a rigid body, there is a great degree of freedom for choosing a body frame for a general rotating and vibrating N-particle system. This, in turn, implies the need to study many-body systems for a class of body frames.The concept of vibrational coordinates and of rotational coordinates is particularly clear for the case of a triatomic molecule: three particles form a single plane, in which the molecule vibrates, and at the same time the molecular plane rotates in the space-fixed frame of reference. The overall molecular rotation is made explicit by means of an orthogonal transformation, which brings the space-fixed into the body-fixed axes. The direction perpendicular to the molecular plane uniquely determines one of the three axes of the frame. Rotation about this direction will change the position of the axes in the molecular plane, in fact the axis system. It is already intuitively clear that this axis transformation will not affect the vibrational portion of the kinetic energy operator. The other parts of the kinetic energy, the rotational part and the rotation-vibration coupling part, will then need to adjust themselves to the new definition of the body frame. In this fashion, we may address the problem of the interaction of rotations and vibrations, which is one of the most important problems in molecular studies.The variation of the rovibrational kinetic energy operator following a change of orientation of the axis system is investigated in the present work from the physical and mathematical points of view. Various types of axis embedding are analyzed in a systematic fashion for triatomic molecules of arbitrary structure, including the bisector, principal axis, and Eckart frames. We also consider the bisector axis embedding for tetratomic molecules and the local axis embedding for pentatomic molecules. Different axis conventions lead sometimes to a complex entanglement of rotational and vibrational motions, which may pose serious obstacles to numerically exact approaches.
We study the quantization of three-dimensional many-body systems in rotating coordinate frames defined implicitly by frame conditions. We carry out the elimination of orientational degrees of freedom in general, giving the Hamiltonian for the NNN-particle system in a broad class of body frames in terms of frame conditions and internal coordinates. We obtain several forms for the kinetic energy operator and compare them to related expressions in the literature.
We study the quantization of three-dimensional many-body systems in rotating coordinate frames defined implicitly by frame conditions. We carry out the elimination of orientational degrees of freedom in general, giving the Hamiltonian for the NNN-particle system in a broad class of body frames in terms of frame conditions and internal coordinates. We obtain several forms for the kinetic energy operator and compare them to related expressions in the literature.
The quantum mechanical two-body problem is analyzed. Separating the center of mass from the relative motion Hamiltonian and focusing on “central potentials,” the stationary Schrödinger equation for the relative motion in spherical coordinates is split into radial and angular equations. The universal angular equation is identified as the eigenvalue equation of the angular momentum operator, whose proper solutions are the spherical harmonics. For fixed interparticle distance, the two-body system is mapped on a “rigid rotor” Hamiltonian, whose eigenstates coincide with the angular momentum eigenstates. In diatomic molecules, timescale separation between fast vibrations (radial motion) and slow rotations (angular motion) enables one to invoke a rigid rotor approximation for interpreting rotational absorption spectrum in the microwave regime. Deviations from the predictions of the rigid rotor model and their manifestation in experiments are analyzed by explicit solution of the stationary Schrödinger equation for two particles in the presence of vibration–rotation coupling.
A several-particle system is called a molecule in the Born–Oppenheimer approximation. The nonrigidity of molecules involves difficulty in molecular dynamics. Guichardet [A. Guichardet, Ann. Inst. H. Poincaré 40, 329 (1984)] showed recently that the vibration motion cannot in general be separated from the rotation motion, by using the connection theory in differential geometry. The point of his theory is the observation that a center-of-mass system is made into a principal fiber bundle with rotation group as the structure group, and is equipped with a connection by the Eckart condition of rotationless constraint. The base manifold of this bundle is called the internal space. The fact that the connection has nonvanishing curvature gives rise to the nonseparability of vibration from rotation. This is a mathematical meaning of nonrigidity of molecules. As an application of the connection theory due to Guichardet, this paper establishes a gauge theory for nonrigid molecules on the basis of the observation that the vector bundle associated with the principal fiber bundle (the center-of-mass system) provides a setting for quantum mechanics of the ‘‘internal’’ molecular motion. The interest, however, centers on planar triatomic molecules in order to put forward the gauge theory in an explicit manner. The conclusion is this: The internal space of a planar triatomic molecule is diffeomorphic with R3−{0}, and endowed with Dirac’s monopole field which may be interpreted as a Coriolis field induced by the rotation. The angular momentum eigenvalues, which are twice the quantized monopole strengths, assign the complex line bundles over the internal space. The internal states of the molecule are described as the cross sections of the complex line bundle, on which the internal Hamiltonian operator acts in minimally coupling with the monopole field.
Abstract The years ago Kramers and pauli quantized the diatomic molecular model which bears their names, obtaining the formula which has placed so large a part in hand spectroscopy. It is sometimes pointed out in modern text-books that these results, founded on the old mechanics, do not offer a very far-reaching analogy to those which the wave mechanics give for the real diatomic molecule. I think the reason will be seen on closely examining the classical calculation, either in the original papers, or in Born's "Atom-mechanik," Without adequate cause, all types of motion save one were rejected in the quantization; all that remained was the simple type of motion which, apart from the gyroscope's axial spin consists merely of a steady precession of the body as a whole about the invariable axis. It will here be shown that in reality we can have Kramers and Pauli models performing a very different type of motion, while completely satisfying the old-quantum equations (which were determined in the original papers). In these motions the gyroscope presesses not only about the invariable axis but also about the nuclear axis. Kemble proposed that in a modified model, having an elastically mounted gyroscope, notions of this more complex type should be possible as quantized States, though his actual treatment is confined to motions of the simpler type. As stated above, we shall see that the original model can have the more complex type of motion, in strictly quantized states. Kemble also proposed an energy formula for the corresponding motions of the real molecule; it is with this and with the well-established molecular formulæ of the wave mechanics that results obtained from the model are to be compared. For specifying orders of magnitude it is convenient to think what happens when the mass of the nuclei is made very great, their angular momentum retaining the same order of magnitude as that of the gyroscope. Then the existence of quantized motions other than those prescribed by Kramers and Pauli is not surprising; for by analogy with the vectorial conception of the real molecular (described, notably, by Hund and Weizel), we should expext, in these circumstances, to find models in whose quantized states the nuclei, if of great mass, will rotate about the invariable axis very s;lowly in comparison with the gyroscope's precession about the nuclear axis. For the real molecule, wave mechanics equations such as Kronig's (22) will hold with an accuracy increasing with the nuclear mass, in the sense that though the main rotational terms will became smaller, the neglected perturbations (if so they may be called) with diminish far more rapidly.
Quantum mechanics for internal motions of the three-body system is set up on the basis of the complex vector bundle theory. The three-body system is called a triatomic molecule in the Born–Oppenheimer approximation. The internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator. This bundle is equipped with a linear connection, which is a natural consequence of a geometric interpretation of the so-called Eckart condition. The coupling of the internal motion with the rotation is understood naturally in terms of this connection. The internal Hamiltonian operator is obtained which includes the internal motion–rotation coupling and a centrifugal potential. The complex vector bundle for the triatomic molecule proves to be a trivial bundle, though the geometric setting for the internal motion is independent of whether the bundle is trivial or not.