A UNIVERSAL ANALYTIC POTENTIAL-ENERGY FUNCTION BASED ON A PHASE FACTOR

Using a field equation with a phase factor, a universal analytic potential-energy function applied to the interactions between diatoms or molecules is derived, and five kinds of potential curves of common shapes are obtained adjusting the phase factors. The linear thermal expansion coefficients and Young's moduli of eleven kinds of face-centered cubic (fcc) metals - Al, Cu, Ag, etc. are calculated using the potential-energy function; the computational results are quite consistent with experimental values. Moreover, an analytic relation between the linear thermal expansion coefficients and Young's moduli of fcc metals is given using the potential-energy function. Finally, the force constants of fifty-five kinds of diatomic moleculars with low excitation state are computed using this theory, and they are quite consistent with RKR (Rydberg-Klein-Rees) experimental values.

Structure and analytical potential energy function for the ground state of the BC^x （x=0, -1）

In this paper, the electronic states of the ground states and dissociation limits of BC and BC- are correctly determined based on group theory and atomic and molecular reaction statics. The equilibrium geometries, harmonic frequencies and dissociation energies of the ground state of BC and BC- are calculated by using density function theory and quadratic CI method including single and double substitutions. The analytical potential energy functions of these states have been fitted with Murrell-Sorbie potential energy function from our ab initio calculation results. The spectroscopic data （αe, ωe and ωeχe） of each state is calculated via the relation between analytical potential energy function and spectroscopic data. All the calculations are in good agreement with the experimental data.

Ab initio calculations of accurate dissociation energy and analytic potential energy function for the second excited state B^1∏ of ^7LiH

The reasonable dissociation limit of the second excited singlet state B1∏ of ^7LiH molecule is obtained. The accurate dissociation energy and equilibrium geometry of the B^∏ state are calculated using a symmetry-adaptedcluster configuration interaction method in full active space. The whole potential energy curve for the B1H state is obtained over the internuclear distance ranging from about 0.10 nm to 0,54 nm, and has a least-square fit to the analytic Murrell-Sorbie function form. The vertical excitation energy is calculated from the ground state to the B^1∏ state and compared with previous theoretical results. The equilibrium internuclear distance obtained by geometry optimization is found to be quite different from that obtained by single-point energy scanning under the same calculation condition. Based on the analytic potential energy function, the harmonic frequency value of the B^1∏ state is estimated. A comparison of the theoretical calculations of dissociation energies, equilibrium interatomic distances and the analytic potential energy function with those obtained by previous theoretical results clearly shows that the present work is more comprehensive and in better agreement with experiments than previous theories, thus it is an improvement on previous theories.

The Analytical Potential Energy Function of NH Radical Molecule in External Electric Field

The geometric structures of an Nit radical in different external electric fields are optimized by using the density functional B3P86/cc-PVSZ method, and the bond lengths, dipole moments, vibration frequencies and IR spectrum are obtained. The potential energy curves are gained by the CCSD （T） method with the same basis set. These results indicate that the physical property parameters and potential energy curves may change with the external electric field, especially in the reverse direction electric field. The potential energy function of zero field is fitted by the Morse potential, and the fitting parameters are in good accordance with the experimental data. The potential energy functions of different external electric fields are fitted adopting the constructed potential model. The fitted critical dissociation electric parameters are shown to be consistent with the numerical calculation, and the relative errors are only 0.27% and 6.61%, hence the constructed model is reliable and accurate. The present results provide an important reference for further study of the molecular spectrum, dynamics and molecular cooling with Stark effect.

Ab initio calculation on accurate analytic potential energy functions and harmonic frequencies of c^3∑g^＋ and B^1-Пu states of dimer 7Li2

The comparison between single-point energy scanning （SPES） and geometry optimization （OPT） in determining the equilibrium geometries of c^3∑g^＋ and B^1-Пu states of dimer 7Li2 is made at numerous basis sets by using a symmetryadapted-cluster configuration-interaztion （SAC-CI） method in the Gaussian 03 program package. In this paper the difference of the equilibrium geometries obtained by SPES and by OPT is reported. The results obtained by SPES are found to be more reasonable than those obtained by OPT in full active space at the present SAC-CI level of theory. And the conclusion is attained that the cc-PVTZ is a most suitable basis set for these states. The calculated dissociation energies and equilibrium geometries are 0.8818 eV and 0.3090 nm for c^3∑g^＋ state, and 0.3668 eV and 0.2932 nm for B^1-Пu state respectively. The potential energy curves are calculated over a wide internuclear distance range from about 2.5α0 to 37α0 and have a least-squares fit into the Murrell-Sorbie function. According to the calculated analytic potential energy functions, the harmonic frequencies （We） and other spectroscopic data （ωeXe, Be and αe） are calculated. Comparison of the theoretical determinations at present work with the experiments and other theories clearly shows that the present work is the most complete effort and thus represents an improvement over previous theoretical results.

Analytical Potential Energy Function,Spectroscopic Constants and Vibrational Levels for A^1E_u^+ State of Dimer ~7Li_2

The symmetry-adapted-duster configuration-interaction method is used to investigate the spectroscopicproperties of ~7Li_2(A^1∑_u^+) over the internuclear distance ranging from 2.4ao to 37ao.The complete potential energycurves are calculated at numbers of basis sets.All the ab initio calculated points are fitted to the analytic MurrellSorbie function and then employed to compute the spectroscopic constants.By comparison,the spectroscopic constantsreproduced by the potential attained at D95(3df,3pd) are found to be very close to the experiments,a^d the values (T_e,D_e,R_e,ω_e,ω_eχ_e,α_e and B_e) are of 1.732 93 eV,1.161 36 eV,0.313 27 nm,251.95 cm^(-1),1.623 cm^(-1),0.005 35 cm^(-1),and0.490 cm^(-1),respectively.With the potential obtained at D95(3df,3pd),the totally 75 vibrational states are found whenJ=0.The vibrational levels,the classical turning points and the inertial rotation constants of the first 68 vibrationalstates are calculated for the first time and compared with the available measurements.Good agreement is obtained.The centrifugal distortion constants of the first 32 vibrational states are also reported for the first time.The reasonabledissociation limit for ~7Li_2(A^1∑_u^+) is deduced using the calculated results at present.

Accurate ab initio-based analytical potential energy function for S_2(~1△_g) via extrapolation to the complete basis set limit

The potential energy curves（PECs） of the first electronic excited state of S2（a^1△g） are calculated employing a multi-reference configuration interaction method with the Davidson correction in combination with a series of correlationconsistent basis sets from Dunning： aug-cc-p VX Z（X = T, Q, 5, 6）. In order to obtain PECs with high accuracy, PECs calculated with aug-cc-p V（Q, 5）Z basis sets are extrapolated to the complete basis set limit. The resulting PECs are then fitted to the analytical potential energy function（APEF） using the extended Hartree–Fock approximate correlation energy method. By utilizing the fitted APEF, accurate and reliable spectroscopic parameters are obtained, which are consistent with both experimental and theoretical results. By solving the Schr o¨dinger equation numerically with the APEFs obtained at the AV6 Z and the extrapolated AV（Q, 5）Z level of theory, we calculate the complete set of vibrational levels, classical turning points, inertial rotation and centrifugal distortion constants.

Machine Learning Kinetic Energy Functional for a One-Dimensional Periodic System

Kinetic energy(KE) functional is crucial to speed up density functional theory calculation. However, deriving it accurately through traditional physics reasoning is challenging. We develop a generally applicable KE functional estimator for a one-dimensional (1D) extended system using a machine learning method. Our end-to-end solution combines the dimensionality reduction method with the Gaussian process regression, and simple scaling method to adapt to various 1D lattices. In addition to reaching chemical accuracy in KE calculation, our estimator also performs well on KE functional derivative prediction. Integrating this machine learning KE functional into the current orbital free density functional theory scheme is able to provide us with expected ground state electron density.

The structure and the potential energy function of AlSO(C_S,X^2A″)

By using the B3P86/aug-cc-pvtz method, the accurate equilibrium geometry of the AlSO （Cs, X2AH） molecule has been calculated and compared with available theoreticM values. The obtained results show that the AlSO molecule has a most stable structure with bond lengths of ROA1= 0.1864 nm, ROS=0.1623 nm, RAIS=0.2450 nm, together with a dissociation energy of 13.88 eV. The possible electronic states and their reasonable dissociation limits for the ground state of the AlSO molecule were determined based on the principle of atomic and molecular reaction statics. The analytic potential energy function of the AlSO molecule was derived by the many-body expansion theory and the contour lines were constructed for the first time, which show the internal information of the AlSO molecule, including the equilibrium structure and stable point. The analysis demonstrates that the obtained potential energy function of AlSO is reaSonable and successful and the present investigations provide important insights for further study on molecular reaction dynamics.

The molecular structure and analytical potential energy function of HCO （X^2A＇）

In this paper the equilibrium structure of HCO has been optimized by using density functional theory （DFT）/ B3P86 method and CC-PVTZ basis. It has a bent （Cs, X^2A＇） ground state structure with an angle of 124.4095 °. The vibronic frequencies and force constants have also been calculated. Based on the principles of atomic and molecular reaction statics, the possible electronic states and reasonable dissociation limits for the ground state of HCO molecule have been determined. The analytic potential energy function of HCO （X^2A＇） molecule has been derived by using the many-body expansion theory. The contour lines are constructed, which show the static properties of HCO （X^2A＇）, such as the equilibrium structure, the lowest energies, etc. The potential energy surface of HCO （X^2A＇） is reasonable and very satisfactory.

The molecular structure and the analytical potential energy function of S2^- and S3^-

In this paper, the equilibrium geometry, harmonic frequency and dissociation energy of S2^- and S3^- have been calculated at QCISD/6-311＋＋G（3d2f） and B3P86/6-311＋＋G（3d2f） level. The S2^- ground state is of 2IIg, the S3^- ground state is of 2B1 and S3^- has a bent （C2v） structure with an angle of 115.65° The results are in good agreement with these reported in other literature. For S3^- ion, the vibration frequencies and the force constants have also been calculated. Base on the general principles of microscopic reversibility, the dissociation limits has been deduced. The Murrell-Sorbie potential energy function for S2^- has been derived according to the ab initio data through the least- squares fitting. The force constants and spectroscopic data for S2^- have been calculated, then compared with other theoretical data. The analytical potential energy function of S3^- have been obtained based on the many-body expansion theory. The structure and energy can correctly reappear on the potential surface.

THE ENERGY FUNCTION WITH RESPECT TO THE ZEROS OF THE EXCEPTIONAL HERMITE POLYNOMIALS

We examine the energy function with respect to the zeros of exceptional Hermite polynomials. The localization of the eigenvalues of the Hessian is given in the general case.In some special arrangements we have a more precise result on the behavior of the energy function. Finally we investigate the energy function with respect to the regular zeros of the exceptional Hermite polynomials.

Investigation of analytical harmonic frequency and potential energy function, vibrational levels for the X^2∑^＋ and A^2Л states of CN radical

This paper calculates the equilibrium structure and the potential energy functions of the ground state （X^2∑^＋） and the low lying excited electronic state （A^2Л） of CN radical are calculated by using CASSCF method. The potential energy curves are obtained by a least square fitting to the modified Murrell-Sorbie function. On the basis of physical theory of potential energy function, harmonic frequency （ωe） and other spectroscopic constants （ωeχe, βe and αe） are calculated by employing the Rydberg-Klei-Rees method. The theoretical calculation results are in excellent agreement with the experimental and other complicated theoretical calculation data. In addition, the eigenvalues of vibrational levels have been calculated by solving the radial one-dimensional SchrSdinger equation of nuclear motion using the algebraic method based on the analytical potential energy function.

Precise calculations on spectroscopic parameters of diatomic molecules using a novel potential energy function

A novel scheme for potential energy functions of diatomic molecules is derived using a function with phase factors. Six kinds of potential curves of common shapes are obtained by adjusting the phase factors. Spectroscopic parameters of the ground states for ten kinds of molecules are calculated using the potential energy functions. The results agree well with experimental data.

The structure and the analytical potential energy function of NH_2(X^2B_1)

This paper reports that the equilibrium structure of NH2 has been optimized at the QCISD/6-311＋＋G （3df, 3pd） level. The ground-state NH2 has a bent （C2v, X^2B1） structure with an angle of 103.0582°. The geometrical structure is in good agreement with the other calculational and experimental results. The harmonic frequencies and the force constants have also been calculated. Based on the group theory and the principle of microscopic reversibility, the dissociation limits of NH2（C2v, X^2B1） have been derived. The potential energy surface of NH2（X^2B1） is reasonable. The contour lines are constructed, the structure and energy of NH2 reappear on the potential energy surface.

A universal potential energy function and precise calculations on the molecular spectra

By using a function with a phase factor,a universal analytic potential energy function applied to the interactions between diatoms or molecules is derived and six kinds of potential curves of common shapes are obtained by adjusting the phase factor.The spectroscopic parameters of ten diatomic molecules are calculated by using the potential energy function;as a consequence,all calculation results are in good agreement with experimental data.

The analytical potential energy function of flue gas SO_2(X^1A_1)

The equilibrium structure of flue gas SO2 is optimized using the density functional theory （DFT）/B3P86 method and CC-PV5Z basis. The result shows that it has a bent （C2v, X1A1） ground state structure with an angle of 119.1184°. The vibronic frequencies and the force constants are also calculated. Based on the principles of atomic and molecular reaction statics （AMIIS）, the possible electronic states and reasonable dissociation limits for the ground state of SO2 molecule are determined. The potential functions of SO and 02 are fitted by the modified Murrell-Sorbie＋c6 （M-S＋c6） potential function and the fitted parameters, the force constants and the spectroscopic constants are obtained, which are all close to the experimental values. The analytic potential energy function of the SO2 （X1A1） molecule is derived using the many-body expansion theory. The contour liues are constructed, which show the static properties of SO2 （XIA1）, such as the equilibrium structure, the lowest energies, the most possible reaction channel, etc.

Risk Assessment for Hybrid AC/DC Systems Based on Transient Energy Function

In order to accurately assess the risk of the hybrid AC/DC network under fault,a risk assessment method for the hybrid AC/DC system based on the transient energy function is proposed.First,based on the energy transfer relationship of the hybrid AC/DC power system,the transient energy function model of the hybrid AC/DC system is established.Based on the operating data of the power grid,the energy function is used as an efficiency variable,and the efficiency variable is integrated into the prior risk probability calculation of nodes in the network,and a Bayesian network-based risk assessment model of hybrid AC/DC system is established.Considering the dynamic update model of network cascading failures,the clique tree propagation algorithm is used to dynamically calculate the posterior risk probability of the node to realize the dynamic assessment of the network risk.Finally,the improved IEEE-39 node hybrid AC/DC system is used as an example for analysis.The results show that the proposed model can not only effectively evaluate the overall safety of the network,but also has feasibility in predicting faults,which can provide a theoretical basis for the stability control of the hybrid AC/DC system.

Analysis of the Energy and Environmental Sustainability of a Built Space System: The Case of Patte d’Oie University Campus in Ouagadougou

The aim of this study was to carry out a dynamic simulation of the energy and environmental performance of a built space system, with a view to assessing its energy and environmental class. The use of a simulation and modeling tool, supported by various methodological references, formed the basis of our approach. Adopting a systemic perspective, we described the structural and functional aspects of the systems making up built spaces, as well as the associated energy flows. Our approach was also based on a typology, taking into account typical days, structural and functional configurations at different scales and angles of observation. The analysis tool we developed in Java was applied to the built space system of the Patte d’Oie university campus in Ouagadougou. Annual electricity consumption was measured at 124387.34 kWh, closely aligned with the average annual electricity bill (125224.31 kWh), with a maximum relative deviation of 1%, followed by a carbon emission balance of 58337.66 kg eq CO2 per year. This validation confirmed the effectiveness of our tool. In addition, following the analysis of electricity consumption using our tool, the university campus was classified in energy class B and environmental class C. These results will be based on the emission factors of the energy mix of the West African Economic and Monetary Union (WAEMU) territory, with particular emphasis on Burkina Faso.

Effect of parallel resonance on the electron energy distribution function in a 60 MHz capacitively coupled plasma

In general,as the radio frequency(RF)power increases in a capacitively coupled plasma(CCP),the power transfer efficiency decreases because the resistance of the CCP decreases.In this work,a parallel resonance circuit is applied to improve the power transfer efficiency at high RF power,and the effect of the parallel resonance on the electron energy distribution function(EEDF)is investigated in a 60 MHz CCP.The CCP consists of a power feed line,the electrodes,and plasma.The reactance of the CCP is positive at 60 MHz and acts like an inductive load.A vacuum variable capacitor(VVC)is connected in parallel with the inductive load,and then the parallel resonance between the VVC and the inductive load can be achieved.As the capacitance of the VVC approaches the parallel resonance condition,the equivalent resistance of the parallel circuit is considerably larger than that without the VVC,and the current flowing through the matching network is greatly reduced.Therefore,the power transfer efficiency of the discharge is improved from 76%,70%,and 68%to 81%,77%,and 76%at RF powers of 100 W,150 W,and 200 W,respectively.At parallel resonance conditions,the electron heating in bulk plasma is enhanced,which cannot be achieved without the VVC even at the higher RF powers.This enhancement of electron heating results in the evolution of the shape of the EEDF from a biMaxwellian distribution to a distribution with the smaller temperature difference between high-energy electrons and low-energy electrons.Due to the parallel resonance effect,the electron density increases by approximately 4%,18%,and 21%at RF powers of 100 W,150 W,and 200 W,respectively.