Claims:1. A processor implemented method, comprising:
obtaining, via one or more hardware processors, one or more input physical parameters corresponding to a multi-element alloy structure (202);
classifying, via the one or more hardware processors, the one or more input physical parameters as one of a first element type, or a second element type to obtain at least one of a first set of elements and a second set of elements respectively (204);
computing, via the one or more hardware processors, one or more of (i) an embedding energy function, and (ii) an atomic electron density for each input physical parameter of the at least one of the first set of elements and the second set of elements (206);
scaling, by using a scaling factor, via the one or more hardware processors, the one or more of (i) the embedding energy function, and (ii) the atomic electron density, computed for each input physical parameter of the at least one of the first set of elements and the second set of elements, to obtain a set of scaled parameters and an elemental interaction pair potential function (208);
identifying, via the one or more hardware processors, one or more dissimilar-type pair interaction potential parameters based on the set of scaled parameters and the elemental interaction pair potential function (210);
sequencing, via the one or more hardware processors, the one or more identified dissimilar-type pair interaction potential parameters and one or more similar-type pair interaction potential parameters to obtain a sequence of the pair type interaction potential parameters (212); and
generating, via the one or more hardware processors, a molecular dynamics potential (MDP) file based on the sequence of the one or more type pair interaction potential parameters (214).

2. The processor implemented method of claim 1, further comprising simulating the MDP file to predict at least one of (i) one or more structural properties, (ii) one or more thermodynamic properties, and (iii) one or more mechanical properties of the multi-element alloy structure.

3. The processor implemented method of claim 1, wherein the first set of elements and the second set of elements are distinct from each other.

4. The processor implemented method of claim 1, wherein the first element type and the second element type are one of a body centered cubic (BCC) element type or a face centered cubic (FCC) element type.

5. The processor implemented method of claim 1, wherein the scaling factor is calculated based on a regression analysis of the atomic electron density at embedding energy function minima and equilibrium electron density values.

6. The processor implemented method of claim 1, wherein the order of the sequence of pair type interaction potential parameters is determined based on a sequence of one or more element types comprised in the one or more input physical parameters.

7. A system (100), comprising:
a memory (102) storing instructions;
one or more communication interfaces (106); and
one or more hardware processors (104) coupled to the memory (102) via the one or more communication interfaces (106), wherein the one or more hardware processors (104) are configured by the instructions to:
obtain one or more input physical parameters corresponding to a multi-element alloy structure;
classify the one or more input physical parameters as one of a first element type, or a second element type to obtain at least one of a first set of elements and a second set of elements respectively;
compute one or more of (i) an embedding energy function, and (ii) an atomic electron density for each input physical parameter of the at least one of the first set of elements and the second set of elements;
scale, by using a scaling factor, the one or more of (i) the embedding energy function, and (ii) the atomic electron density, computed for each input physical parameter of the at least one of the first set of elements and the second set of elements, to obtain a set of scaled parameters and an elemental interaction pair potential function;
identify one or more dissimilar-type pair interaction potential parameters based on the set of scaled parameters and the elemental interaction pair potential function;
sequence the one or more identified dissimilar-type pair interaction potential parameters and one or more similar-type pair interaction potential parameters to obtain a sequence of the pair type interaction potential parameters; and
generate a molecular dynamics potential (MDP) file based on the sequence of the one or more type pair interaction potential parameters.

8. The system of claim 6, wherein the one or more hardware processors are further configured by the instructions to simulate the MDP file to predict at least one of (i) one or more structural properties, (ii) one or more thermodynamic properties, and (iii) one or more mechanical properties of the multi-element alloy structure.

9. The system of claim 6, wherein the first set of elements and the second set of elements are distinct from each other.

10. The system of claim 6, wherein the first element type and the second element type are one of a body centered cubic (BCC) element type or a face centered cubic (FCC) element type.

11. The system of claim 6, wherein the scaling factor is calculated based on a regression analysis of the atomic electron density at embedding energy function minima and equilibrium electron density values.

12. The system of claim 6, wherein the order of the sequence of pair type interaction potential parameters is determined based on a sequence of one or more element types comprised in the one or more input physical parameters.
, Description:FORM 2

THE PATENTS ACT, 1970
(39 of 1970)
&
THE PATENT RULES, 2003

COMPLETE SPECIFICATION
(See Section 10 and Rule 13)

Title of invention:
GENERATING MOLECULAR DYNAMICS POTENTIALS AND SIMULATING THEREOF FOR PREDICTING PROPERTIES OF MULTI-ELEMENT ALLOY STRUCTURES

Applicant
Tata Consultancy Services Limited
A company Incorporated in India under the Companies Act, 1956
Having address:
Nirmal Building, 9th Floor,
Nariman Point, Mumbai 400021,
Maharashtra, India

The following specification particularly describes the invention and the manner in which it is to be performed.
TECHNICAL FIELD
The disclosure herein generally relates to simulation techniques for multi-element alloy structures, and, more particularly, to generating molecular dynamics potentials and simulating thereof for predicting properties of multi-element alloy structures.

BACKGROUND
Alloys are widely used because of its superiority in terms of strength, toughness, thermal and other mechanical properties than its constituent elements. It is difficult to determine structural, thermodynamic, and mechanical properties of new alloys precisely because alloys do not just follow predictable rule of mixture of constituting element properties. As a result, much costly and time-consuming experimental testing are required in the development and investigating properties of new metallic alloys.
Classical molecular dynamics (MD) simulations have become ubiquitous tool for simulating various nanoscopic to macroscopic properties in many complex alloy systems such as local relaxed structure, thermodynamic changes, short-range ordering, dislocation movement and strengthening effect. One of the major issues, which limits the ability for materials scientists to design, develop and process metallic materials from their atomic structure using classical MD, is the lack of accurate and suitable interatomic molecular dynamics potentials (MDPs) for the targeted alloy system. Interatomic potentials describe interactions between atoms of alloy systems using computationally efficient functions to solve the atomic movements. These potentials can range from simple pair interactions to more complex formulations that involve the local atomic electron density, bonds, angles, and torsion. Of these, the embedded-atom method (EAM) has been used widely because it can accurately model thermodynamic properties, stability of the structure and defects in metals. Although many good-quality EAM potentials exists for a considerable number of elemental metals, suitable potentials for alloy systems are sparsely available in literature. This makes fast computational exploration of new alloy compositions difficult.

SUMMARY
Embodiments of the present disclosure present technological improvements as solutions to one or more of the above-mentioned technical problems recognized by the inventors in conventional systems. For example, in one aspect, there is provided a processor implemented method for generating molecular dynamics potentials and simulating thereof for predicting properties of multi-element alloy structures. The method comprises obtaining, via one or more hardware processors, one or more input physical parameters corresponding to a multi-element alloy structure; classifying, via the one or more hardware processors, the one or more input physical parameters as one of a first element type, or a second element type to obtain at least one of a first set of elements and a second set of elements respectively; computing, via the one or more hardware processors, one or more of (i) an embedding energy function, and (ii) an atomic electron density for each input physical parameter of the at least one of the first set of elements and the second set of elements; scaling, by using a scaling factor, via the one or more hardware processors, the one or more of (i) the embedding energy function, and (ii) the atomic electron density, computed for each input physical parameter of the at least one of the first set of elements and the second set of elements, to obtain a set of scaled parameters and an elemental interaction pair potential function; identifying, via the one or more hardware processors, one or more dissimilar-type pair interaction potential parameters based on the set of scaled parameters and the elemental interaction pair potential function; sequencing, via the one or more hardware processors, the one or more identified dissimilar-type pair interaction potential parameters and one or more similar-type pair interaction potential parameters to obtain a sequence of the pair type interaction potential parameters; and generating, via the one or more hardware processors, a molecular dynamics potential (MDP) file based on the sequence of the one or more type pair interaction potential parameters. In an embodiment, the order of the sequence of pair type interaction potential parameters is determined based on a sequence of one or more element types comprised in the one or more input physical parameters.
In an embodiment, the method further comprises simulating the MDP file to predict at least one of (i) one or more structural properties, (ii) one or more thermodynamic properties, and (iii) one or more mechanical properties of the multi-element alloy structure.
In an embodiment, the first set of elements and the second set of elements are distinct from each other.
In another embodiment, the first element type and the second element type are one of a body centered cubic (BCC) element type or a face centered cubic (FCC) element type.
In an embodiment, the scaling factor is calculated based on a regression analysis of the atomic electron density at embedding energy function minima and equilibrium electron density values.
In another aspect, there is provided a system for generating molecular dynamics potentials and simulating thereof for predicting properties of multi-element alloy structures. The system comprises: a memory storing instructions; one or more communication interfaces; and one or more hardware processors coupled to the memory via the one or more communication interfaces, wherein the one or more hardware processors are configured by the instructions to: obtain one or more input physical parameters corresponding to a multi-element alloy structure; classify the one or more input physical parameters as one of a first element type, or a second element type to obtain at least one of a first set of elements and a second set of elements respectively; compute one or more of (i) an embedding energy function, and (ii) an atomic electron density for each input physical parameter of the at least one of the first set of elements and the second set of elements; scale, by using a scaling factor, the one or more of (i) the embedding energy function, and (ii) the atomic electron density, computed for each input physical parameter of the at least one of the first set of elements and the second set of elements, to obtain a set of scaled parameters and an elemental interaction pair potential function; identify one or more dissimilar-type pair interaction potential parameters based on the set of scaled parameters and the elemental interaction pair potential function; sequence the one or more identified dissimilar-type pair interaction potential parameters and one or more similar-type pair interaction potential parameters to obtain a sequence of the pair type interaction potential parameters; and generate a molecular dynamics potential (MDP) file based on the sequence of the one or more type pair interaction potential parameters.
In an embodiment, the one or more hardware processors are further configured by the instructions to simulate the MDP file to predict at least one of (i) one or more structural properties, (ii) one or more thermodynamic properties, and (iii) one or more mechanical properties of the multi-element alloy structure.
In an embodiment, the first set of elements and the second set of elements are distinct from each other.
In another embodiment, the first element type and the second element type are one of a body centered cubic (BCC) element type or a face centered cubic (FCC) element type.
In an embodiment, the scaling factor is calculated based on a regression analysis of the atomic electron density at embedding energy function minima and equilibrium electron density values.
In yet another aspect, there are provided one or more non-transitory machine-readable information storage mediums comprising one or more instructions which when executed by one or more hardware processors cause a method for generating molecular dynamics potentials and simulating thereof for predicting properties of multi-element alloy structures. The method comprises obtaining, via one or more hardware processors, one or more input physical parameters corresponding to a multi-element alloy structure; classifying, via the one or more hardware processors, the one or more input physical parameters as one of a first element type, or a second element type to obtain at least one of a first set of elements and a second set of elements respectively; computing, via the one or more hardware processors, one or more of (i) an embedding energy function, and (ii) an atomic electron density for each input physical parameter of the at least one of the first set of elements and the second set of elements; scaling, by using a scaling factor, via the one or more hardware processors, the one or more of (i) the embedding energy function, and (ii) the atomic electron density, computed for each input physical parameter of the at least one of the first set of elements and the second set of elements, to obtain a set of scaled parameters, and an elemental interaction pair potential function; identifying, via the one or more hardware processors, one or more dissimilar-type pair interaction potential parameters based on the set of scaled parameters and the elemental interaction pair potential function; sequencing, via the one or more hardware processors, the one or more identified dissimilar-type pair interaction potential parameters and one or more similar-type pair interaction potential parameters to obtain a sequence of the pair type interaction potential parameters; and generating, via the one or more hardware processors, a molecular dynamics potential (MDP) file based on the sequence of the one or more type pair interaction potential parameters. In an embodiment, the order of the sequence of pair type interaction potential parameters is determined based on a sequence of one or more element types comprised in the one or more input physical parameters.
In an embodiment, the method further comprises simulating the MDP file to predict at least one of (i) one or more structural properties, (ii) one or more thermodynamic properties, and (iii) one or more mechanical properties of the multi-element alloy structure.
In an embodiment, the first set of elements and the second set of elements are distinct from each other.
In another embodiment, the first element type and the second element type are one of a body centered cubic (BCC) element type or a face centered cubic (FCC) element type.
In an embodiment, the scaling factor is calculated based on a regression analysis of the atomic electron density at embedding energy function minima and equilibrium electron density values.
It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS
The accompanying drawings, which are incorporated in and constitute a part of this disclosure, illustrate exemplary embodiments and, together with the description, serve to explain the disclosed principles:
FIG. 1 illustrates an exemplary block diagram of a system for generating molecular dynamics potentials and simulating thereof for predicting properties of multi-element alloy structures, in accordance with an embodiment of the present disclosure.
FIG. 2 illustrates an exemplary flow diagram of a method for generating molecular dynamics potentials and simulating thereof for predicting properties of multi-element alloy structures using the system of FIG. 1 in accordance with an embodiment of the present disclosure.
FIG. 3 depicts a graphical representation of a comparison and scaling of atomic electron density values at the Embedding Energy minima for a body centered cubic (BCC) element type and/or a face centered cubic (FCC) element type molecular dynamics potential (MDP) framework, using the system of FIG. 1, in accordance with an embodiment of the present disclosure.
FIG. 4 depicts a graphical representation of a comparison and scaling of atomic electron density values at the next neighbor atomic distance for the body centered cubic (BCC) element type and/or the face centered cubic (FCC) element type MDP framework, using the system of FIG. 1, in accordance with an embodiment of the present disclosure.
FIG. 5 depicts a graphical representation of an elemental interaction pair potential function, in accordance with an embodiment of the present disclosure.
FIG. 6 depicts a graphical representation illustrating a comparison of Embedding energy minima at equilibrium atomic electron density (?e) of elements calculated by BCC and FCC MDP framework, using the system of FIG. 1, in accordance with an embodiment of the present disclosure.

DETAILED DESCRIPTION OF EMBODIMENTS
Exemplary embodiments are described with reference to the accompanying drawings. In the figures, the left-most digit(s) of a reference number identifies the figure in which the reference number first appears. Wherever convenient, the same reference numbers are used throughout the drawings to refer to the same or like parts. While examples and features of disclosed principles are described herein, modifications, adaptations, and other implementations are possible without departing from the scope of the disclosed embodiments.
Metallic alloys are known for their superior properties in terms of enhanced strength and toughness at ambient and elevated temperatures which is applicable for many structural and applications. Traditionally, the new alloy development and alloy processing involved various high-end expansive experiments, development time for 10-20 years and cost of required man-hours. Nowadays with the rise in computation power, molecular dynamics (MD) simulation has become a major
computational tool, which simulates metallic alloy systems for their nanostructure, thermodynamic and mechanical properties by involving molecular dynamics potentials (MDP). This MD based simulations can reduce the number of required experiments and subsequently significantly reduce the cost of alloy development and speed up the development works. However, one of the major issues, which limits the ability for materials scientists to design metallic materials from the atoms using classical Molecular Dynamics, is the lack of accurate interatomic molecular dynamics potentials (MDPs). Suitable MDPs of the desired alloy systems makes fast exploration of new alloy compositions and related properties, which is very difficult and time-consuming process otherwise through experimental routes. The present disclosure provides systems and methods for developing the essential molecular dynamics potential for new or traditional metallic alloys for their computer simulated structural, thermodynamic, and mechanical property predictions. More specifically, embodiments of the present disclosure provide systems and methods for generating MDP for multi-element alloy systems wherein both Body Centered Cubic (BCC) element type and/or a Face Centered Cubic (FCC) element type are combined. Pure elements and multi-element alloys of combinations of BCC and FCC elements are modeled for predicting their various structural, thermodynamic, and mechanical properties.
Referring now to the drawings, and more particularly to FIG. 1 through 6, where similar reference characters denote corresponding features consistently throughout the figures, there are shown preferred embodiments and these embodiments are described in the context of the following exemplary system and/or method.
FIG. 1 illustrates an exemplary block diagram of a system for generating molecular dynamics potentials and simulating thereof for predicting properties of multi-element alloy structures, in accordance with an embodiment of the present disclosure. In an embodiment, the system 100 includes one or more processors 104, communication interface device(s) or input/output (I/O) interface(s) 106, and one or more data storage devices or memory 102 operatively coupled to the one or more processors 104. The one or more processors 104 may be one or more software processing modules and/or hardware processors. In an embodiment, the hardware processors can be implemented as one or more microprocessors, microcomputers, microcontrollers, digital signal processors, central processing units, state machines, logic circuitries, and/or any devices that manipulate signals based on operational instructions. Among other capabilities, the processor(s) is configured to fetch and execute computer-readable instructions stored in the memory. In an embodiment, the device 100 can be implemented in a variety of computing systems, such as laptop computers, notebooks, hand-held devices, workstations, mainframe computers, servers, a network cloud and the like.
The I/O interface device(s) 106 can include a variety of software and hardware interfaces, for example, a web interface, a graphical user interface, and the like and can facilitate multiple communications within a wide variety of networks N/W and protocol types, including wired networks, for example, LAN, cable, etc., and wireless networks, such as WLAN, cellular, or satellite. In an embodiment, the I/O interface device(s) can include one or more ports for connecting a number of devices to one another or to another server.
The memory 102 may include any computer-readable medium known in the art including, for example, volatile memory, such as static random access memory (SRAM) and dynamic random access memory (DRAM), and/or non-volatile memory, such as read only memory (ROM), erasable programmable ROM, flash memories, hard disks, optical disks, and magnetic tapes. In an embodiment a database 108 can be stored in the memory 102, wherein the database 108 may comprise, but are not limited to information pertaining to physical parameters corresponding to a multi-element alloy structure, output(s) generated by one or more simulation technique(s), one or more modeling technique(s), scaling techniques, scaling factors, etc. In an embodiment, the memory 102 may store the one or more modeling technique(s), the one or more simulation technique(s), which are execcuted by the one or more hardware processors 104 to perform the methodology described herein. The memory 102 further comprises (or may further comprise) information pertaining to input(s)/output(s) of each step performed by the systems and methods of the present disclosure. In other words, input(s) fed at each step and output(s) generated at each step are comprised in the memory 102 and can be utilized in further processing and analysis.
FIG. 2, with reference to FIG. 1, illustrates an exemplary flow diagram of a method for generating molecular dynamics potentials and simulating thereof for predicting properties of multi-element alloy structures using the system 100 of FIG. 1, in accordance with an embodiment of the present disclosure. In an embodiment, the system(s) 100 comprises one or more data storage devices or the memory 102 operatively coupled to the one or more hardware processors 104 and is configured to store instructions for execution of steps of the method by the one or more processors 104. The steps of the method of the present disclosure will now be explained with reference to the components of the system 100 as depicted in FIG. 1, and the flow diagram. In an embodiment of the present disclosure, at step 202, the one or more hardware processors 104 obtain one or more input physical parameters corresponding to a multi-element alloy structure. In an embodiment, the one or more input physical parameters comprise lattice parameter, cohesive energy, vacancy formation energy, second order elastic constants, and the like. Table 1 and Table 2 depict the one or more input physical parameters corresponding to the multi-element alloy structure (e.g., say Nickel-Vanadium (NiV or Ni-V) alloy structure, palladium-manganese (PdMn or Pd-Mn) alloy structure.
Table 1
Metal a (Å) Ec (eV) Ef (eV) C11(eV) C12(eV) C44(eV)
Ta 3.3025 9.71 2.95 1.648 0.986 0.516
Mo 3.147 8.691 3.1 3.017 0.954 0.786
W 3.165 11.070 3.95 3.263 1.267 0.9987
Fe 2.866 5.479 1.79 1.342 0.89 0.557

Table 2

Parameters Fe Mo Ta W Ni Pd
re 2.481 2.728 2.860 2.740 2.488 2.750
fe 1.885 2.723 3.086 3.487 2.007 1.595
pe 20.041 29.354 33.787 37.234 27.562 21.335
ps 20.041 29.354 33.787 37.234 27.930 21.940
a 9.818 8.393 8.489 8.900 8.383 8.697
ß 5.236 4.476 4.527 4.746 4.471 4.638
A 0.392 0.708 0.611 0.882 0.429 0.406
B 0.646 1.120 1.032 1.394 0.633 0.599
? 0.170 0.137 0.176 0.139 0.443 0.397
? 0.340 0.275 0.353 0.278 0.820 0.754
Fn0 -2.534 -3.692 -5.103 -4.946 -2.693 -2.321
Fn1 -0.059 -0.178 -0.405 -0.148 -0.076 -0.473
Fn2 0.193 0.380 1.112 0.365 0.241 1.615
Fn3 -2.282 -3.133 -3.585 -4.432 -2.375 -0.231
F0 -2.540 -3.710 -5.140 -4.960 -2.700 -2.360
F1 0.000 0.000 0.000 0.000 0.000 0.000
F2
0.200 0.875 1.640 0.661 0.265 1.481
F3 -0.148 0.776 0.221 0.348 -0.152 -1.675
? 0.391 0.790 0.848 -0.582 0.469 1.130
Fe -2.539 -3.712 -5.141 -4.961 -2.699 -2.352

At step 202 of the present disclosure, the one or more hardware processors 104 classify the one or more input physical parameters as one of a first element type, or a second element type to obtain at least one of a first set of elements and a second set of elements respectively. The first set of elements and the second set of elements are distinct from each other. For example, the first element type and the second element type are one of a body centered cubic (BCC) element type or a face centered cubic (FCC) element type. It is to be understood by a person having ordinary skill in the art or person skilled in the art that input physical parameters pertaining to BCC and FCC elements shall not be construed as limiting the scope of the present disclosure. More specifically, Table 1 depicts the input physical parameters of BCC elements and Table 2 depicts the input physical parameters of FCC elements both of which are obtained from experimental or theoretical data, in one example embodiment of the present disclosure. In the experiments conducted by the present disclosure, two different suitable EAM MDP alloy development methodologies were chosen. The first EAM MDP methodology is only reserved for BCC elements. The second EAM MDP methodology is developed predominantly for FCC elements. But in the second EAM MDP methodology, there is also a provision for developing MDP for some BCC elements.
Referring to steps of FIG. 2, at step 206 of the present disclosure, the one or more hardware processors 104 compute one or more of (i) an embedding energy function, and (ii) an atomic electron density for each input physical parameter of the at least one of the first set of elements and the second set of elements. Input parameters such as lattice constant, cohesive energy, vacancy formation energy and other elastic constants for the models are taken from experimental values or Density Functional Theory (DFT) calculations available in the literature. Using the mathematical formulation of different methodologies for BCC and FCC type elements, MDP component tables of embedding energy, atomic electron density and pairwise interaction potential between dissimilar elements were determined/computed for all different elements. Below description illustrates computation of one or more of (i) an embedding energy function, (ii) an atomic electron density, and (iii) an elemental interaction pair potential function for BCC elements.
An analytical nearest neighbor potential for BCC element-based alloys is developed first for a few BCC elements, in one example embodiment of the present disclosure. The basic governing equations of EAM model as described earlier are:
E_total= ?¦??F(??_i)+1/2 ?_(i,j(i?j))¦??F(r?_ij)??
?_i= ?_(j?i)¦??f(r?_ij)?
In order to apply the model, the embedding function F(?), atomic electron density function f(r), and the two-body pair interaction potential function F(r) must be known. The atomic electron density around each BCC atom is given by:
fr = fe (r1e/r)ß
where fe is a factor which is determined by
fe = Ec/?
Ec is the cohesive energy, O is the atomic volume, r1e is the equilibrium first neighbor distance, and ß is an adjustable parameter taken as ß=6 for all of the BCC alloy systems.
F(?) = -(Ec - Ef)((1-ln?(?/?e))n) (?/?e)n
Here Ef is the unrelaxed vacancy formation energy, ? is the equilibrium atomic electron density, and n is a parameter that is given by:
n=1/ß [(9OB-15OG)/(Ec - Ef)]^(1/2)
where B is the bulk modulus and G is the Voigt average shear modulus.
The monoatomic/ single-element pair interaction potential is taken as cubic spline function and is illustrated as below expression:
F (r)=k_3 [r/r_1e -1]^3+k_2 [r/r_1e -1]^2+ k_1 [r/r_1e -1]^1+k_0
where, k constants depend upon a combination of elastic constants and elastic anisotropy.
However, the pair interaction potential of two dissimilar type BCC elements, a and b is given as alloy potential is:
F ab=1/2 [fb(r)/fa(r) Faa+fa(r)/fb(r) Fbb]
where, F aa and F bb are monoatomic potentials given by monoatomic models. fa(r) and fb(r) are atomic electron density function for a- and b-type atoms.
Below description illustrates computation of one or more of (i) an embedding energy function, (ii) an atomic electron density, and (iii) an elemental interaction pair potential function for FCC elements.
In this EAM alloy potential model, the generalized elemental pair potentials are expressed as:
F (r)= Aexp[-a(r/re-1)]/(1+(r/re-k)^20 )- Bexp[-ß(r/re-1)]/(1+(r/re-?)^20 )
where re is the equilibrium spacing between nearest neighbors, A, B, a, and ß are four adjustable parameters, and k and ? are two additional parameters for the cutoff distance of interaction. The electron density function around each atom is taken as:
f (r)= (f_e exp?[-ß(r/re-1)])/(1+(r/re-?)^20 )
Embedding energy functions that works well over a wide range of atomic electron density require that three equations be used to separately fit three different atomic electron density ranges. For a smooth variation of the embedding energy, these equations are required to match values and slopes at their junctions.
To have embedding energy functions that can work well over a wide range of atomic electron density ?, we have used three equations to separately fit to different atomic electron density ranges: ? < ?n, ?n< ?< ?o and ?o < ?. By using ?n = 0.85?e and ?o = 1.15?e where ?e is the equilibrium electron density, it is ensured that all equilibrium properties can be fitted in the atomic electron density range: ?n< ?< ?o.
These equations are given by:
F(?)= ?_(i=0)^3¦F_ni (?/(?_n )-1)^i,? < ?_n,? ??_n=0.85?_e
F(?)= ?_(i=0)^3¦F_i (?/(?_e )-1)^i, ?_n< ?< ?_0, ?_0 =1.15?_e
F(?)=F_e [1-?ln?{?/?_s }?^n ] {?/?_s }^n,? ??_0