Effective torsion-rotation Hamiltonian for methanol-type molecules

A fourth-order effective Hamiltonian has been derived for the torsion-rotation problem of a methanol-type molecule, i.e., for a C_3v top attached to a C_s frame. First, symmetry considerations based on a frame-fixed axis system are used to determine allowed terms in the Hamiltonian. These terms are then subjected to a contact transformation to remove the indeterminate ones. This procedure is essentially an extension of Watson's method for semirigid molecules to the torsion-rotation problem. It is demonstrated that the Hamiltonian derived in the present work is capable of improving the fittings of the millimeter and submillimeter absorption frequencies of CH₃OH and CH₃SH.     

The large-amplitude motions in quasi-symmetric top molecules with internal C3v rotors: The vibration-torsion-rotation Hamiltonian of a two-top molecule

An expression for the vibration-torsion-rotation Hamiltonian of a molecule with triatomic nonrigid frame and two internal C_3v rotors has been derived. Three large-amplitude motions, namely skeletal bending and two torsions, are removed from the vibrational problem and are considered together with the rotational problem. The Hamiltonian obtained is applicable to any two-C_3v-top molecule with triatomic linear or bent frame. The zeroth-order skeletal bending-torsion-rotation Hamiltonian is derived and the method of solving the corresponding Schrödinger equation is discussed. The Hamiltonian obtained with only slight modifications is shown to be applicable to any single-C_3v-top molecule with a quasi-linear tetratomic nonrigid frame or to the problem of the large-amplitude bending motion in a pentatomic quasi-linear molecule.     

Critical phenomena in the rotational spectra of H2X molecules

Using the method of generating functions for the effective rotational Hamiltonian of H₂X molecules, an expression for estimating the critical value J _c of the rotational quantum number is obtained, starting with which the formation of four-level clusters in a molecule is possible. The analysis made in the harmonic approximation for rotational operators and for a particular form of generating functions showed the impossibility of clusterization for nonrigid H₂X molecules with strong centrifugal effects.     

Theory of an Fm3m→I4/m structural phase transition in an Rb2KScF6 crystal

An Rb₂KScF₆ crystal having an elpasolite structure undergoes a sequence of Fm3m→I4/m→P12₁/n1 structural phase transitions where the transition to the tetragonal phase is associated with “rotation” of the ScF₆ octahedron. An effective Hamiltonian is constructed to describe the Fm3m→I4/m transition using the approximation of a local mode for which we selected a “soft mode” whose eigenvector corresponds to the rotation of the octahedron. The effective Hamiltonian also includes the relationship between the local mode and the homogeneous elastic strains. The parameters of the effective Hamiltonian were determined using the generalized Gordon-Kim model of an ionic crystal which allows for the deformability and polarizability of the ions. The thermodynamic properties of a system with this model Hamiltonian were investigated using the Monte Carlo method. The calculated phase transition temperature of 250 K is almost the same as the experimental value (252 K). The tetragonal phase remains stable as far as T=0 K and a second transition (to the monoclinic phase) cannot be obtained using this effective Hamiltonian. This suggests that if the transition to the tetragonal phase is mainly associated with “rotations” of the octahedrons, in order to describe the phase transition to the monoclinic phase the effective Hamiltonian must allow for additional degrees of freedom mainly associated with the motion of rubidium ions.     

Transformation of the effective rotational Hamiltonian of nonrigid X 2 Y molecules

The transformation of the effective rotational Hamiltonian H of nonrigid X ₂ Y molecules to the form having a minimum number of diagonals in the basis of rotational functions of a symmetric top is discussed. Such a transformation is a generalization of the reduction transformation performed for the polynomial effective Hamiltonian H. It is shown that in the general case the transformation substantially changes the form of the initial Hamiltonian, which restricts the region of applicability (J<J*) of the reduced Hamiltonian represented in a class of elementary functions in terms of angular momentum operators. The values of the rotational quantum number J* are estimated for the (000) ground and (010) vibrational states of the H₂O molecule.     

Derivation of the nonrigid rotation-large-amplitude internal motion Hamiltonian of the general molecule

The nonrigid (effective) rotation-large-amplitude internal motion Hamiltonian (NRLH) of the general molecule with one or more large-amplitude vibrations has been derived to the order of magnitude κ²T_VIB. The derivation takes advantage of the idea of a nonrigid reference configuration and uses the contact transformation method as a mathematical tool. The NRLH has a form fairly similar to that of the effective rotation Hamiltonian of semirigid (i.e., normal) molecules. From a careful examination of the Eckart-Sayvetz conditions and of the Taylor expansions of the potential energy surface in terms of curvilinear displacement coordinates, three types of large-amplitude internal coordinates of different physical meaning (effective large-amplitude internal coordinates, real large-amplitude internal coordinates, and reaction path coordinates) are described. To test the ideas and the formulas the effective bending potential function of the C₃ molecule in its ground electronic and ground stretching vibrational state is calculated from the ab initio potential energy surface given by W. P. Kraemer, P. R. Bunker, and M. Yoshimine (J. Mol. Spectrosc. 107, 191–207 (1984)). The calculations were carried out by using either the effective or the real large-amplitude bending coordinate of C₃. The NRLH theory is compared to the nonrigid bender theory at a theoretical level as well as through the results of the test calculations.     

A reduced form of the spin-rotation Hamiltonian for asymmetric-top molecules,with applications to HO2 and NH2

The spin-rotational Hamiltonian for an asymmetric-top molecule in a given vibrational level of an open-shell electronic state may contain more parameters than can be determined from the observed energy levels. This paper describes the reduction of the Hamiltonian by means of a unitary transformation to a form suitable for fitting to observed energies. It is shown that, for molecules of lower than orthorhombic symmetry, there are fewer determinable quadratic spin-rotation parameters than have been used previously. For example, for a molecule belonging to the group C₈, there are four, not five, determinable spin-rotation constants, ?_αβ. Similar indeterminacies exist among the quartic terms of the spin-rotation Hamiltonian. The case of a molecule of orthorhombic symmetry, for which there are six determinable quartic parameters, is considered in detail. The results are applied to the experimental data available on the spin-rotation splittings of the HO₂ and NH₂ radicals in their ground vibrational and electronic states.     

Magnetocaloric features of complex molecular magnets: The (Cr7Ni)2Cu molecular magnet and beyond

We study the new kind of systems represented by the Cr₇Ni-M-Cr₇Ni (M=Cu⁺²) molecule, which is a promising molecular achievement from the perspective of molecular electronics. By using an effective quantum Hamiltonian, an exact calculation of the magnetic specific heat C_Mag and the magnetocaloric features, namely, the adiabatic change of the entropy ΔS_Mag and temperature ΔT_ad, respectively, are developed. A systematic simulation of the magnetocaloric properties is generated by modifying the effective exchange couplings into the molecular system. Extended discussion of calculated magnetocaloric features and its possible realization by experimental methods, are performed. In addition, comparisons with an exact numerical result and with a Van Vleck transformation, which has important application in similar micromagnetic structures with no exact analytical solution and larger Hilbert space, are presented. Moreover, an expression for the entangling-excitation frequencies of these systems is given as first application of our simplified solution to the effective molecular Hamiltonian.     

Vibrational relaxation of a molecule in strong interaction with a reservoir: Nonmonotonic temperature dependence

This theoretical study of the vibrational relaxation of a molecule in interaction with a reservoir uncovers a noteworthy temperature (T) dependence of the time evolution of the relaxation. Its rate increases with T in one interval but decreases in another. The feature arises not for a weak molecule-reservoir interaction but only for coupling strong enough to require polaronic dressing transformations. Our treatment, based on a recent generalization of the well-known Montroll–Shuler equation for relaxation and an explicit calculation of bath correlations from the microscopically specified Hamiltonian, could provide an alternative explanation of an “inverted” T-dependence of relaxation in an experimental report by Fayer and collaborators on W(CO)₆ dissolved in CHCl₃.     

Analysis of the microwave spectrum of hydrazine

Microwave measurements in the interval from 6 to 133 GHz, consisting of 444 rotational transitions in the vibrational ground state of hydrazine with J ≤ 31 and K_a ≤ 6 were fit to an effective rotational Hamiltonian containing 9 asymmetric rotor constants, 14 NH₂ inversion parameters, and 1 internal rotation parameter, with an overall standard deviation of the fit of 0.40 MHz. This set of parameters contains: (i) the three rotational constants; (ii) tunneling splitting constants for NH₂ inversion at one end of the molecule, for NH₂ inversion at both ends of the molecule, and for internal rotation through the trans barrier; (iii) two K-type doubling constants affecting the K = 1 levels; (iv) an a-type Coriolis interaction with matrix elements linear in K; and (v) various centrifugal distortion corrections to the above parameters. A consistent group theoretical formalism was used to label the energy levels and to select terms in the phenomenological rotational Hamiltonian. The Hamiltonian matrix, which is set up in a tunneling basis set, is of dimension 16×16 and contains only ΔK_a = 0 matrix elements, asymmetric rotor effects being taken into account on the diagonal by terms from a Polo expansion in bⁿ. Hyperfine splittings and barrier heights are not discussed.     

Centrifugal distortion coefficients of asymmetric-top molecules: Reduction of the octic terms of the rotational Hamiltonian

The rotational Hamiltonian of an asymmetric-top molecule in its standard form, containing terms up to eighth degree in the components of the total angular momentum, is transformed by a unitary transformation with parameters S_pqr to a reduced Hamiltonian so as to avoid the indeterminacies inherent in fitting the complete Hamiltonian to observed energy levels. Expressions are given for the nine determinable combinations of octic constants Θ′_i (i = 1 to 9) which are invariant under the unitary transformation. A method of reduction suitable for energy calculations by matrix diagonalization is considered. The relations between the coefficients of the transformed Hamiltonian, for suitable choice of the parameters S_pqr, and those of the reduced Hamiltonian are given. This enables the determination of the nine octic constants Θ′_i in terms of the experimental constants.     

The vibration-rotation problem in triatomic molecules allowing for two large-amplitude stretching vibrations

An expression for the Hamiltonian of a vibration-rotating triatomic molecule is derived, using two curvilinear stretching coordinates ?₁ and ?₃ and one rectilinear bending coordinate S₂, in such a way that the Hamiltonian obtained is applicable to any bent triatomic molecule and allows for large displacements along the stretching coordinates. From this, a zeroth-order Hamiltonian H_s⁰ (?₁, ?₃) is obtained, describing the energy levels associated with the two stretching vibrations ν₁ and ν₃. The vibrational energy levels (v₁, v₃^even) of an XY₂ molecule having unequal bond lengths at equilibrium are then calculated. The kinetic energy T₀ (?₁, ?₃) of the Hamiltonian effectively takes into account the two large-amplitude motions in ν₁ and ν₃ together with their interaction. A model calculation is described for a bent XY₂ molecule (SO₂ in its ¹A′ (¹B₂) excited state) in which the ν₃ oscillation occurs in a double-minimum potential. Coupling by kinetic energy terms in the Hamiltonian turns out to be very small in this example.     

The development of a new Morse-oscillator based rotation-vibration Hamiltonian for H3+

The rotation-vibration Hamiltonian for an equilateral triangular X₃ molecule is derived in terms of the three curvilinear stretching coordinates Δr_i, and expanded in the form of a power series in the variables y_i = 1 ? exp(-aΔr_i), where a is a molecular parameter obtained from the potential function. The reason for the use of the variable y_i is twofold: Stretching potentials exhibit a much stronger convergence in the y_i than in the Δr_i, and a Hamiltonian expressed in the y_i can be diagonalized in a straightforward fashion using a Morse-oscillator basis set as we do here. Using a published ab initio potential surface we have expanded it as a polynomial in the y_i, and have calculated variationally the rotation-vibration energies of H₃⁺ and D₃⁺ using a symmetry-adapted Morse-oscillator-rigid symmetric top basis set. The results indicate that an expansion of the potential function to quartic terms in the y_i might be adequate, and that satisfactorily converged energies can be obtained with a relatively small basis set. The molecule H₃⁺ is the simplest polyatomic molecule. Inspection of the Appendix of this paper shows that the rotation-vibration Hamiltonian used here is one of the more complicated ones.     

The two-dimensional anharmonic oscillator: Vibration-rotation constants of fulminic acid,HCNO

The lowest-wavenumber vibration of HCNO and DCNO, ν₅, is known to involve a largeamplitude low-frequency anharmonic bending of the CH bond against the CNO frame. In this paper the anomalous vibrational dependence of the observed rotational constants B(v₅, l₅), and of the observed l-doubling interactions, is interpreted according to a simple effective vibration-rotation Hamiltonian in which the appropriate vibrational operators are averaged in an anharmonic potential surface over the normal coordinates (Q_5x, Q_5y). All of the data on both isotopes are interpreted according to a single potential surface having a minimum energy at a slightly bent configuration of the HCN angle (～170°) with a maximum at the linear configuration about 2 cm^?1 higher. The other coefficients in the Hamiltonian are also interpreted in terms of the structure and the harmonic and anharmonic force fields; the substitution structure at the “hypothetical linear configuration” determined in this way gives a CH bond length of 1.060 Å, in contrast to the value 1.027 Å determined from the ground-state rotational constants.We also discuss the difficulties in rationalizing our effective Hamiltonian in terms of more fundamental theory, as well as the success and limitations of its use in practice.     

On the barriers to planarity and the isotope effect in cyclobutane and cyclobutane-d8

The infrared and Raman data on the ring-puckering vibration in cyclobutane and cyclobutane-d₈ have been reexamined including the coordinate dependence of the reduced mass in the Hamiltonian. This was done for the purpose of estimating the importance of these small terms in the determination of barrier heights for four-membered rings and also on the determination of the dihedral angle corresponding to the potential minimum.The conclusions reached are that there is an isotopic dependence of the barriers to planarity in cyclobutane and cyclobutane-d₈ yielding a difference of ～14 cm^?1, but the precise value of the difference in barrier heights is ill determined. The higher-order kinetic energy terms in the Hamiltonian can account for a spread of ～3 cm^?1 in each of the barriers derived for cyclobutane and cyclobutane-d₈, depending on the details of the model used for the vibration, but not a difference of 14 cm^?1, which undoubtedly indicates the effects of coupling with other vibrational modes. It is also found that the derived values of the dihedral angles are quite sensitive to the details of the vibrational model, in fact, much more so than to the uncertainties in the bond distances and bond angles. A relationship between the potential constants derived for cyclobutane and cyclobutane-d₈ assuming an effective constant reduced mass and those derived for a semirigid model is demonstrated.     

The rotational dependence of the Renner-Teller interaction: a new term in the effective Hamiltonian for linear triatomic molecules in Π electronic states

An additional term in the effective Hamiltonian for a molecule in a ^2S+1Π state subject to a small Renner—Teller effect is derived. It takes the form of a rotational dependence of the Renner—Teller operator and is assigned the parameter ?ω_2,D . This term reproduces two characteristics of the spin—rotational levels of Renner—Teller molecules, both of which are well documented by experimental examples but are not explicable in terms of the previous effective Hamiltonian. These are the small difference between the B values of different vibronic components of the same bending vibrational level and the anomalously strong dependence of the spin—rotation parameter γ on ν₂ for molecules with S > 0. These effects have been explained by previous workers using perturbation theory but in a less general and more complicated fashion.     

Inversion and internal rotation in methyl-d-amine: Theory

A theory has been developed for an analysis of the microwave spectrum of the CH₂DNH₂-type molecule which has an asymmetric internal rotor. First, the Hamiltonian matrix was expressed on the basis of localized wavefunctions, each of which corresponds to a conformer vibrating in the vicinity of a potential minimum. Next, by a symmetrization, the Hamiltonian matrix was factored into four submatrices. By solving these matrices, a general view of the energy-level structure has been given, which should be useful for an interpretation of the observed rotational spectrum. It has been shown that the inversion splitting in each level of the trans form molecule should be sensitive to the amount of trans-gauche coupling through tunneling and therefore the relative height of a trans level with respect to a gauche level can be determined from an observation of the inversion splitting in the trans levels.     

Rotation-vibration energy levels of H2O and C3 calculated using the nonrigid bender Hamiltonian

We have written a new computer program for diagonalizing the nonrigid bender Hamiltonian, and have based the program entirely on the theory as reviewed by P. Jensen [Comp. Phys. Rep. 1, 1–56 (1983)] and P. Jensen and P. R. Bunker [J. Mol. Spectrosc. 118, 18–39 (1986)]. Using this program we can calculate the rotation-vibration energy levels of a triatomic molecule from the potential energy function. The program is an improvement over an earlier version, particularly in the systematic treatment of all singular terms, and in the allowance made for the dependence of all perturbation energy denominators on the bending quantum number v₂ and rotation quantum number K. The new program can be used for symmetric and unsymmetric triatomic molecules. In the present paper we test the program by applying it to the calculation of the rotation-vibration energy levels of C₃ from an ab initio potential surface, and of H₂O from ab initio and experimental potential surfaces.     

Vibrational anharmonicity and the inversion potential function of NH3

The nonrigid bender formulation of the vibration-inversion-rotation Hamiltonian for an XY₃ pyramidal molecule (V. ?pirko, J. M. R. Stone, and D. Papou?ek, J. Mol. Spectrosc.60, 159 (1976)) is improved by allowing for anharmonicity in all the vibrations. To model the anharmonic potential function of an XY₃ molecule with a low barrier to inversion a Plíva-type empirical potential (J. Plíva, Collect. Czech. Chem. Commun.23, 777 (1958)) is used. A fitting procedure that involves the numerical integration of the effective inversion Schrödinger equation (the nonrigid bender equation) and diagonalization of some resonance matrices is used to determine the equilibrium structure and the anharmonic potential function of the ammonia molecule.