A microscopic calculation of force constants in crystals

A microscopic theory has been developed which splits up the total electronic valence charge density into partial densities (PD's) that can unambiguously be assigned to specific ion cores. These PD's together with the ion cores constitute the basic building entities of the crystal whose interacting forces can easily be obtained. This enables one to calculate the interatomic force constants in harmonic approximation also. Further by dividing the displacement induced electronic charge density variation into a rigid shift and an additional distortion contribution the influence of these special degrees of freedom on the force constants in real space can be studied separately. The method is presented using Si as an example.     

On the force constants of graphene

It has been shown that the inclusion of an additional short-range potential in the repulsion energy of neighboring atoms allows one to satisfactorily describe the frequency of the phonon doublet at the Γ point for atomic vibrations in the plane of the graphene sheet and to calculate the corresponding Grüneisen coefficient. The results obtained are compared with those derived by other authors.     

Ab initio calculation of force constants and equilibrium geometries

Force constants of the molecules HF, NH₃, CH₄ and BH₄ ^- have been calculated ab initio by the force method with a 73/3 + 1 gaussian lobe basis set. The results, including a former calculation on H₂O, agree well with experiment: the average relative error is 12 per cent for the diagonal force constants and the average absolute error is 0·06 mdyn/Å for the off-diagonal ones. The trends are also correctly reproduced. It is concluded that ab initio calculations of this accuracy can help to solve a number of spectroscopic problems. Force constants of BH₄ ^- have been determined from a combination of spectroscopic and ab initio information. Geometries have been obtained with little computing work and show good agreement with experiment.     

On the calculation and interpretation of centrifugal distortion constants: A statistical basis for model testing: The calculation of the force field

Watson's treatment of the theory of centrifugal distortion effects is reviewed in detail and special attention is drawn to the relationship between Watson's treatment and the Kivelson-Wilson treatment. The relationship between the distortion parameters and the force field for XYZ-type molecules is also reviewed in order to remove ambiguities and inconsistencies presently found in the literature. Finally, procedures for exploiting the statistics associated with a least-squares analysis of the data are presented as a guide for model testing and for detecting misassignments and mismeasurements. A series of examples illustrating the results of these procedures are presented using the published spectra of SO₂, OF₂, SiF₂ and NSF. In the cases of OF₂, SiF₂ and NSF, improved values for the rotational constants are obtained.     

Quantum-chemical calculation of force constants and electrooptical parameters of mono-α-chloroethers

Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 55, No. 2, pp. 251–257, August, 1991.     

Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules

The force constants and equilibrium geometry of water were calculated ab initio from Hartree-Fock self-consistent field wavefunctions using the force method, and a 53/31, a 73/3 + 1 and a 95/41 + 2 gaussian lobe basis set. The force method proved to be very economic and numerically accurate.

The calculated values agree well with the experimental ones. Especially good is the agreement for the interaction constant F_rα . Calculated force constants for the 95/41 + 2 basis are: F_r = 9·158 mdyn/å, F_α = 0·8513 mdyn/å, F_rα = +0·3007, F_rr = -0·1724; the experimental values from isotope frequencies are: F_r = 8·456, F_α = 0·762, F_rα = +0·246, F_rr = -0·100 mdyn/å. The agreement in the geometry is also good for the 95/41 + 2 basis r _e = 0·9518 å, α = 107·49°, and the experimental r _e = 0·9572 å, α = 104·52°.

Force constants were found to be quite insensitive to variation in the basis functions, all three sets giving almost the same result. On the other hand, the dipole moment derivative is very sensitive.     

The application of the matrix method of successive approximations to the calculation of the force constants of tetramethylmethane

A study is made of the applicability of the matrix method of successive approximations to the calculation of force constants under conditions of insufficient experimentally measured vibrational frequencies using as an example the molecule tetramethylmethane and its deuteron modifications. A solution is guaranteed by fixing some of the force constants in each approximation. A procedure is proposed for the selection of the force constants to be fixed, and the assignment of values to them. The force constants obtained in this way completely satisfactorily reproduce the experimental vibration frequencies.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 7, pp. 59–62, July, 1971.     

On the calculation of force constants using isotopic frequencies by parametric method: Application to systems of (3×3) order

A method is proposed for the determination of angle parameters for ONF, ONCl and ONBr, utilising the isotopic frequencies of two isotopically substituted molecules. The force constants, Coriolis coupling constants, inertia defect, mean amplitudes of vibration and rotational distortion constants were also calculated and compared with literature values.     

Adiabatic calculation of the three-nucleon force

A preliminary estimate of the three-nucleon force within the adiabatic framework has already been done. In this work we try to improve the first calculation by using relativistic kinematics for the pion and including theP 11 absorption channel, as well as theP 33 and the less importantS 13,S 11,P 13,P 31 channels, in theN interaction.Presented at the symposium Mesons and Light Nuclei, Bechyn, Czechoslovakia. May 27–June 1, 1985.     

On the linear chain with arbitrary masses and force constants. I. Correlation functions

Simple expressions are derived for the time dependent correlation functions of certain phase functions of the classical linear (harmonic) chain having arbitrary masses and force constants. This is done for a statistical distribution τ (H, P) in phase space depending arbitrarily on the energy H and the total momentum P, which then is specialized to the uniform distributions τ_H, on the surface of constant energy, and τ_H,P, in the intersection of surfaces of constant energy and total momentum. In a subsequent paper²) the results will be used to draw conclusions on the ergodic properties of the phase functions.     

On the linear chain with arbitrary masses and force constants. II. Ergodic properties

The ergodic properties of linear and quadratic phase functions of the classical linear chain are studied for the uniform statistical distributions on the energy surface and in the manifold belonging to fixed values of the energy and the total momentum. This is done for the finite chain by using the time dependent correlation functions studied in a previous paper¹). The thermodynamic limit is also discussed. As an example, sufficient conditions on the masses and force constants are given to ensure that the kinetic energy of a certain particle remains nonergodic in the thermodynamic limit, the conditions defining a non-exceptional set of chains.     

Calculation of molecular geometries and force constants from CNDO wavefunctions by the force method

The semi-empirical quantum chemical calculation of molecular geometries and force constants by the usual energy hypersurface method rapidly becomes impractical as the size of the molecule increases. Here we show that the application of the force method to semi-empirical wavefunctions makes an economic and simple calculation of molecular geometries and force constants possible. Problems which are virtually insoluble by the classical method, such as full geometry optimization in large molecules, can be solved this way.

The calculation of forces as exact negative derivatives of the total SCF energy is given for CNDO wavefunctions. Two geometry optimization schemes are discussed ; it is concluded that a steepest descent method is the most practical in semi-empirical calculations. As an example the fully optimized equilibrium geometry of pyrrole has been determined. Except for the CH and NH bond lengths, the calculated geometry agrees almost perfectly with the experimental one.

The calculation of force constants from the forces is discussed. The force constants of CH₄ and H₂O have been determined, including the interaction ones. The signs and magnitudes of the stretch-bend interaction constants agree with experiment.