Embedded atom method (EAM) potential with user defined functions

This example demonstrates the use of the embedded atom method (EAM) potential routine together with user defined functions. The potential form and parameters have been taken from [MisMehPap01]. The example also illustrates the use of XML Inclusions. Note that the potential is merely evaluated for a couple of simple lattice structures and none of the parameters are fitted.

Location

examples/potential_EAM

Input files

  • main.xml: main input file

     1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
    <?xml version='1.0' encoding='iso-8859-1'?>
<job>

  <name>Fitting an EAM potential for copper</name>
  
  <verbosity>medium</verbosity>
  
  <fitting>
    <BFGS conv-threshold='1e-5' max-iter='20' />
  </fitting>
  
  <atom-types>
    <species>Cu</species>
  </atom-types>
  
  <potentials>
    <xi:include href='potential.xml'
		xmlns:xi='http://www.w3.org/2003/XInclude' />
  </potentials>
  
  <structures>
    <xi:include href='structures.xml'
		xmlns:xi='http://www.w3.org/2003/XInclude' />
  </structures>

</job>

  • potential.xml: initial parameter set (included in main input file via XML Inclusions)

      1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
    <eam id="elemental EAM potential" species-a="*" species-b="*">

  <mapping>
    <pair-interaction species-a="*" species-b="*" function="V" />
    <electron-density species-a="*" species-b="*" function="rho" />
    <embedding-energy species="*" function="F" />			
  </mapping>

  <functions>

    <sum id="V">
      <user-function id="V_term1">
	<input-var>r</input-var>
	<expression>
	  ( E1 * (exp(-2 * alpha1 * (r - r01))
	  - 2 * exp(-alpha1 * (r - r01)))
	  + E2 * (exp(-2 * alpha2 * (r - r02))
	  - 2 * exp(-alpha2 * (r - r02))) + delta) 
	</expression>
	<derivative>
	  E1 * (2 * alpha1 * exp(-alpha1 * (r - r01))
	  - 2 * alpha1 * exp(-2 * alpha1 * (r - r01))) 
	  + E2 * (2 * alpha2 * exp(-alpha2 * (r - r02))
	  - 2 * alpha2 * exp(-2 * alpha2 * (r - r02)))
	</derivative>
	<param name="E1">2.01458e2</param>
	<param name="alpha1">2.97758</param>
	<param name="r01">0.83591</param>
	<param name="E2">6.59288e-3</param>
	<param name="alpha2">1.54927</param>
	<param name="r02">4.46867</param>
	<param name="delta">0.86225e-2</param>
	<screening>
	  <user-function id="rho_screening">
	    <cutoff>5.50679</cutoff>
	    <input-var>r</input-var>
	    <expression>
	      1 - 1/(1 + ((r - cutoff) / h)^4)
	    </expression>
	    <derivative>
	      4 * h^4 * (r-cutoff)^3 / ((h^4 + (r-cutoff)^4)^2)
	    </derivative>
	    <param name="h">0.50037</param>
	  </user-function>
	</screening>
      </user-function>
      <user-function id="V_term2">
	<cutoff>5.50679</cutoff>
	<input-var>r</input-var>
	<expression>
	  - (r &lt; rs1 ? S1*(rs1-r)^4 : 0)
	  - (r &lt; rs2 ? S2*(rs2-r)^4 : 0)
	  - (r &lt; rs3 ? S3*(rs3-r)^4 : 0)
	</expression>
	<derivative>
	  - (r &lt; rs1 ? -4*S1*(rs1-r)^3 : 0)
	  - (r &lt; rs2 ? -4*S2*(rs2-r)^3 : 0)
	  - (r &lt; rs3 ? -4*S3*(rs3-r)^3 : 0)
	</derivative>
	<param name="rs1">2.24</param>
	<param name="rs2">1.80</param>
	<param name="rs3">1.20</param>
	<param name="S1">4.0</param>
	<param name="S2">40.0</param>
	<param name="S3">1.15e3</param>
      </user-function>
    </sum>

    <user-function id="rho">
      <input-var>r</input-var>
      <expression>
	a * exp(-beta1*(r - r03)^2)
        + exp(-beta2*(r - r04))
      </expression>
      <derivative>
        beta2 * (-exp(-beta1 * (r - r04)))
	- 2 * a * beta1 * (r - r03) * exp(-beta1 * (r - r03)^2)
      </derivative>
      <param name="a">3.80362</param>
      <param name="r03">-2.19885</param>
      <param name="r04">-2.61984e2</param>
      <param name="beta1">0.17394</param>
      <param name="beta2">5.35661e2</param>
      <screening>
	<user-function id="rho_screening">
	  <cutoff>5.50679</cutoff>
	  <input-var>r</input-var>
	  <expression>
	    1 - 1/(1 + ((r - cutoff) / h)^4)
	  </expression>
	  <derivative>
	    4 * h^4 * (r - cutoff)^3 / ((h^4 + (r - cutoff)^4)^2)
	  </derivative>
	  <param name="h">0.5</param>
	</user-function>
      </screening>
    </user-function>

    <user-function id="F">
      <input-var>rho</input-var>
      <expression>
	(rho &lt; 1) ?
	(F0 + 0.5 * F2 * (rho - 1)^2 + q1 * (rho - 1)^3
	+ q2 * (rho - 1)^4 + q3 * (rho - 1)^5 + q4 * (rho - 1)^6)
	: 
	(F0 + 0.5 * F2 * (rho - 1)^2 + q1 * (rho - 1)^3
	+ Q1 * (rho - 1)^4) / (1 + Q2 * (rho - 1)^3)
      </expression>
      <derivative>
	(rho &lt; 1) ?
	(rho - 1) * ( F2
	+ 3 * q1 * (rho - 1)
	+ 4 * q2 * (rho - 1)^2
	+ 5 * q3 * (rho - 1)^3
	+ 6 * q4 * (rho - 1)^4 )
	:
	(F2 * (rho - 1) + 3 * q1 * (rho - 1)^2
	+ 4 * Q1 * (rho - 1)^3)/(Q2 * (rho - 1)^3 + 1)
	- (3 * Q2 * (rho - 1)^2 * (F0 + 0.5 * F2 * (rho - 1)^2
	+ q1 * (rho - 1)^3 + Q1 * (rho - 1)^4))
	/ ((Q2 * (rho - 1)^3 + 1)^2)
      </derivative>
      <param name="F0">-2.3</param>
      <param name="F2">1.4</param>
      <param name="q1">-1.3</param>
      <param name="q2">-0.9</param>
      <param name="q3">1.8</param>
      <param name="q4">3.0</param>
      <param name="Q1">0.4</param>
      <param name="Q2">0.3</param>
    </user-function>

  </functions>

</eam>

  • structures.xml: input structures (included in main input file via XML Inclusions)

     1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
    <group>

  <fcc-lattice id='FCC'>
    <atom-type>Cu</atom-type>
    <lattice-parameter>3.615</lattice-parameter>
    <properties>
      <atomic-energy target='-3.540'/>
      <lattice-parameter target='3.615'/>
      <bulk-modulus/>
      <C11 target='170.0'/>
      <C12 target='122.5'/>
      <C44 target= '75.8'/>
    </properties>
  </fcc-lattice>

  <bcc-lattice id='BCC'>
    <atom-type>Cu</atom-type>
    <lattice-parameter>2.8</lattice-parameter>
    <properties>
      <atomic-energy target='-3.496'/>
      <lattice-parameter/>
    </properties>
  </bcc-lattice>

  <diamond-lattice id='DIA'>
    <atom-type>Cu</atom-type>
    <lattice-parameter>4.0</lattice-parameter>
    <properties>
      <atomic-energy target='-2.293'/>
      <lattice-parameter min='3.0'/>
    </properties>
  </diamond-lattice>	

  <sc-lattice id='SC'>
    <atom-type>Cu</atom-type>
    <lattice-parameter>2.3</lattice-parameter>
    <properties>
      <atomic-energy target='-2.996'/>
      <lattice-parameter/>
    </properties>
  </sc-lattice>	

</group>

Output (files)

  • The final properties (as well as parameters) are written to standard output.

     1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
    This is program version 0.1.6
Reading job file main.xml
-------------------------------------------------------
Parsing input file(s)
-------------------------------------------------------
-------------------------------------------------------
Potential parameters being optimized (#dof=0):
   NONE
-------------------------------------------------------
-------------------------------------------------------
Number of properties being fitted: 0
-------------------------------------------------------
-------------------------------------------------------
Computing structure properties
Structure 'FCC':
   total-energy: -3.55787 eV
   atomic-energy: -3.55787 eV/atom
   total-volume: 11.8104 A^3
   atomic-volume: 11.8104 A^3/atom
   bulk-modulus: 139.856 GPa
   C11: 171.427 GPa
   C12: 124.138 GPa
   C44: 76.1944 GPa
   lattice-parameter: 3.615 A [0:]
Structure 'BCC':
   total-energy: -3.4849 eV
   atomic-energy: -3.4849 eV/atom
   total-volume: 10.976 A^3
   atomic-volume: 10.976 A^3/atom
   lattice-parameter: 2.8 A [0:]
Structure 'DIA':
   total-energy: 7.19803 eV
   atomic-energy: 3.59901 eV/atom
   total-volume: 16 A^3
   atomic-volume: 8 A^3/atom
   lattice-parameter: 4 A [3:]
Structure 'SC':
   total-energy: -3.04011 eV
   atomic-energy: -3.04011 eV/atom
   total-volume: 12.167 A^3
   atomic-volume: 12.167 A^3/atom
   lattice-parameter: 2.3 A [0:]
-------------------------------------------------------
-------------------------------------------------------