A Novel Method for Calculation of Molecular Energies and Charge Distributions by Thermodynamic Formalization

The paper describes a new approach to the thermodynamic formalization for calculation of molecular energy and charge distribution in ground state by means of the variational equation of DFT. In order to thermodynamically formalize the molecular calculation, the pseudo chemical potential (PCP) is conceptualized, where a molecule is broken into multi-phase(atom) one-component(electron) systems and the energy of system is represented as PCP. Calculation of the molecular energy and atomic charge by PCP is put forward, thereafter the approach is proved to be valid and its efficiency (accuracy and calculation speed) is verified.

In quantum theory for calculating molecular energy, atomic charge, structures and properties, excellent successes were achieved in calculation for many projects by establishment to ab initio and semi-empirical method of wave function theory (WFT) [1][2][3][4][5][6] in that electron wave function is variable quantity of energy minimization and density-function theory (DFT) 7,8 in that electron density is variable quantity.
Of course, the ab initio method of WFT is widely used for calculating the properties of the molecules because of its high accuracy [9][10][11][12][13] . However, it is infeasible for calculating the molecules with a large quantity of atoms, because the more quantity of atoms is given, the longer it takes to calculate. Therefore, less accurate DFT based on electron density has been created and used [14][15][16][17][18][19][20] . Over the past decades, much effort has been made to improve the calculation accuracy of DFT [7][8][9][10][11][12][13][14][15]21,22 . WFT, DFT and its varieties are all different in calculation accuracy and velocity, but both formalization of the variable quantity of energy minimization and the solution of the variational equation are based on the quantum theoretical formalization.
The aim of this paper is to lay foundations for the principles and methods which could determine the energy of molecules and their atomic charges by thermodynamic formalization, rather than the quantum theory.
In the variational equation which gives the electron density corresponding to minimum energy in DFT, Euler-Lagrange (EL) equation, an undefined multiplier μ is the energy per electron and electron migration measure.
(μ: chemical potential, E: the energy of an electron system, N: number of electrons, v r ( ): external potential). This μ can be also defined in the ground state of a pure molecule equal to temperature zero limit state of finite temperature thermal equilibrium system. That is where θ is finite temperature of the thermal equilibrium system, A is the free energy of θ v r , ( ) electron system and S is entropy of electron system under θ v r , ( ). A pure state molecule electron system, not a mixed one, can be supposed to be a thermodynamic system. In this regard, the state density and the state sum depend on the energy level and degeneracy in formalization of the universally used statistical models of electron system (statistical model 23 of atom, statistical model 24 of 2-atom molecule, gas model [25][26][27][28][29][30] of free electron, Thomas-Fermi model 31,32 of DFT), and from the statistical viewpoint, the molecule can be regarded as the thermodynamic system, based on the fact that the property of the macroscopic system, thermodynamic property, can be obtained by means of the state sum.
From such viewpoint, to applicate the method of thermodynamic formalization, we taken undefined multiplier μ in the EL equation as a function form proportional to the logarithm of the electron number and then defined it as the pseudo chemical potential (PCP). We divided the total energy, the quantum mechanical average value of molecule Hamilton operator, into atom sections and proposed a new molecule energy calculation model which is described as a PCP. Hence, the molecule energy is denoted by the function of electron number and electron number is made to become the variable quantity of energy minimization rather than wave function or electron density. The partial derivative value of the α-th atom's electron number for the total molecular energy is defined and formalized as a pseudo chemical potential of each phase (atom) (PCPP).
On the basis of the above definition, we proposed the thermodynamic model of multi-phase one component molecular system, in which atoms are regarded as phase and electrons are regarded as component. In addition, variation principle is put forward, which means the molecule formation process is the minimization process of energy and equalizing process of each phase's PCPP as well.
The molecule is formed when PCPP of each phase is equally formed. From thermodynamic viewpoint, it is similar to the Gibbs' phase equilibrium condition where the chemical potentials of each phase and component are equal. From the PCPP equilibrium condition, the simultaneous equation (variational equation) where the electron number N is an undefined number is obtained. The number of electrons, the solution of simultaneous equation, is a stationary value that gives the minimum value of energy in the ground state of the molecule and the value that determines the atom charge in the ground state as well. The thermodynamic formalization has been evaluated through the calculation experiment. What is unique in this method is that a new thermodynamic model for molecular electron system is proposed and calculating system for the molecular system is established by using the method of thermodynamic formalization. To the best of our knowledge, in previous researches, there has been no research that described the energies of the molecular electron systems as chemical potentials and the charge distribution of atoms in molecules is realized by the method of the thermodynamic formalization.

theoretical foundation
Definition of PCP by thermodynamic method. EL Eq. (3) 7,8,31 of DFT is obtained by variation of energy of electronic system (6) under constraint (7).
As mentioned above, for the sake of thermodynamic formulation, the undetermined multiplier µ is defined as the function in proportion to logarithm of electron number, hereafter called PCP, which is formulated as follows: where N 0 is electron number of neutral atom, N is electron number of composed atom of molecule, µ 0 is µ value when N = N 0 , γ is parameter.
where c is integration constant, γ has dimension of energy. Integration constant is conditionally set as zero because it is related to reference point. γ is obtained from energy of ionization (I) and electron affinity (A) and Eq. (9) as follows Model of total energy calculation of a molecule. For a start, according to the PCP concept, the electrons are localized into atoms (domain) so that the molecules become the set of atoms with different number of electrons, in order to apply the thermodynamic formalization method by making molecules multi-phase one-component system where the localized atoms are deemed as phase and the electrons as component in temperature zero limit thermal equilibrium system. In Born-Oppenheimer approximation, the total energy of molecules Hamiltonian is given as follows: e n e e e n n To construct multi-phase one-component system model, the electron coordinates are replaced by nucleus ones by setting distance between electrons as − ≈ The nucleus-electron interaction energy in molecule V ne is the sum of interaction terms of V ne,a in domains or those of domains.
where Z α is charge of α atom, N β is electron number of β atom domain, V ne,a is nucleus-electron interaction energy in α atom domain. If density matrix is approximated to ρ (r 1 , r 2 ) ≈ ρ (r 1 ) ρ (r 2 ), electron-electron interaction energy is also the sum of two term.
The nucleus-nucleus interaction term is Therefore, from Eqs. 15~18, total energy allotted to atom domain is represented as follows. N Z N Z , the total energy of molecule E is given as follows, where the sum of the total energy in α atom domain is represented by the sum of interaction term between E α and atom domains; (20) equation to calculate molecular energy by means of pcp. Total energy of molecule is approximately calculated, but it is based on quantum chemical average value of total energy operator. Therefore, total energy of molecule is represented by sum of energy per atom domain and electrostatic interaction energy between domains. By substituting Eq. (10) for Eq. (20), total energy of molecule, following Eq. is obtained.
where E M has no property to calculate the quantum chemical average value of operator, but the thermodynamic formulization only. In addition, E M cannot be assumed to be the total energy of molecule like Eq. (20). Because PCP is energy to each electron numerically, it is not an absolute value of energy but a variation proportion. E M in Eq. (21) is defined as molecular energy and used as initial equation. Regarding a pure-state molecule as a multi-phase one-component system demands complicate approximation, which means it would not be suitable for quantitative calculation. To obtain a correct value, correction factor (parameter) k αβ is applied.
where k αβ is an interaction parameter between α atom and β atom.
Parameter k αβ of Eq. (22) is successfully estimated, and database is constructed.
pseudo chemical potential of each phase (atom) (pcpp) in molecule. First, according to the common method in thermodynamics, pseudo chemical potential of each (atom) phase (PCPP) µ α M that composes molecules is determined by calculating partial derivative to Eq. 23.
where µ α M is the PCPP of α-th phase and the function of N and R αβ as well. How to form a molecule from atoms is interpreted by variable principle to variable quantity N.
The formation process of the molecules from atoms is a minimizing process of energy by migration of electrons between the phases (atoms), that is the equalizing process of µ α M . Because state of minimum energy is a temperature zero limit thermoequilibrium state, in the closed system, could be obtained.
According to the principle that charge or number of particles are also conserved in molecules as in the closed thermodynamic system, the following equation must be given;

Results and Discussion
Efficiency of PCP method. Total energy of some molecules calculated by PCP method and their calculation duration is shown in Table 1. Comparing against other methods, this method is fast in calculation speed and high in accuracy. The calculation duration gets longer when the atoms in the molecules are big in number. For instance, in case of the ab initio method and DFT method, when each of them is given more than 130 and 30 atoms respectively, it is infeasible due to their prolonged calculation duration. Furthermore, the results of this method are very similar to the one of ab initio method and DFT method, indicating that it is higher in accuracy. evaluation of accuracy. To further evaluate its accuracy, it is compared to the ab initio method, calculating the total energies of some organic and inorganic molecules, as shown in Table 2 and Table 3, respectively. The results in their Tables clearly show that calculation results of the new method are approximate to one of ab initio methods, and on the other hand, the relative errors in calculating value of all organic and inorganic molecules are rare (lower than 0.0176), indicating that PCP method is accurate in calculation of total energy of the molecules. Figure 1 shows the correlation analysis of the total energies of some molecules, organic (Fig. 1a) and inorganic (Fig. 1b). Regression equations are expressed as follows, respectively:

conclusions
In summary, we have proposed a novel method for calculation of the molecular electron system by thermodynamic formalization rather than quantum chemical method.
The         Table 3. Calculation results of total energy of some inorganic molecules.   Thereafter, the function values of each atom can be calculated; At this time, the inter-nucleus distances are calculated from molecular structure data obtained by molecular-dynamics method as follows; On the other hand, according to the atomic charge calculated by using the method above, total energy of the molecule can be calculated as follows; In addition, PCPs of the molecules are also calculated in the calculation process.