Energies of Isolated Atoms

The ΔHf calculated by semiempirical methods is defined as the energy in kcal.mol-1 required to form one mole of the system in the gas phase at 298K from its elements in their standard state:

begin{displaymath}Delta H_f = E_{elect} + E_{nuc} + sum_AE_{isol}(A) +sum_AE_{atom}(A)end{displaymath}

In order to calculate ΔHf , the quantity Eisol must be determined; this is the energy required to form the isolated atom from its valence electrons:

begin{displaymath}E_{isol}(A)=E_{rm neutral atom}(A) -E_{rm nucleus}(A)-E_{rm valence electrons}(A)end{displaymath}

In the calculation of Eelect, the energy of valence electrons is defined as zero, likewise in calculating Enuc, the energy of the isolated nucleus is defined as zero, therefore the calculation of Eisol simplifies to the calculation of Eneutral atom.

The energy of Eeisol is the energy released when the valence electrons are added to the nucleus. For example, for the hydrogen atom, this would be Uss. For poly-electronic atoms, the electron-electron interactions must be included, in addition to the one-electron contributions. Most elements have open shell ground states, and for these systems, the nature of the state is important.

For all main group elements, that is, elements with valence shell configurations of the form nsanpb, other than the alkali metals, the value of Eisol is given by:

Eisol =aUss+bUpp+(a-1)Gss+a.bGsp+(b(b-1))/2Gp2-bHsp- cHpp

in which c=min(b(b-1)/2,(6-b).(5-b)/2). Except for the Hpp term, all the contributions to Eisol are obvious. Non-zero Hpp terms occur when there are two or more unpaired electrons in the ground state, in which case there is an exchange stabilization that is otherwise absent.

Because Hpp is usually written as 1/2(Gpp-Gp2), the expression for systems with 2 to 4 p electrons is recast as:

Eisol =aUss+bUpp+(a-1)Gss+a.bGsp+((b(b-1))/2+c/2)Gp2- (a-1)bHsp-c/2Gpp,

or

Eisol =2Uss+bUpp+Gss+2.bGsp+((b(b-1))/2+c/2)Gp2- (a-1)bHsp-c/2Gpp.

For the alkali metals, the equation for Eisol is the same as that for hydrogen.

For the transition metals, the coefficients for the d-d interactions are more complicated.

The general form for Eisol for a transition metal of configuration smdns, in which there are ma α s-electrons and mb β s-electrons, and na α d-electrons and nb β d-electrons, and the total angular quantum number is L, is:

Eisol

=

mUss+nUdd+ (m(m-1))/2Gss+m.nGsd-(mana+mbnb)Hsd

 

 

 

 

As might be imagined, derivation of this expression is by no means obvious, particularly the terms for Gdd2 and Gdd4. Interested readers are referred to Racah's paper in Phys Rev, 61, 186 (1942). In this, Racah derived an expression for the d orbital energy of the ground state in terms of three quantities, A, B, and L, the total angular momentum:

The quantities A and B, and a third quantity, C, not used here, are related to the Gk as follows:

A

=

Gdd0-49Gdd2

B

=

Gdd2-5Gdd4

C

=

35Gdd4

 

 

 

Using Racah's equation, derivation of Eisol is straightforward. In texts on transition metal ion theory, the quantities Gdd0, Gdd0, and Gdd0 are usually represented by the symbols F0, F2, and F4, respectively. However, care should be exercised when reading these texts: sometimes other quantities, F0, F2, and F4 are used. The relationship between these three sets of symbols is as follows:

Gdd0

=

F0

=

F0

Gdd2

=

F2

=

F2/49

Gdd4

=

F4

=

F4/441

 

 

 

 

 

Because the coefficients for the two electron terms are so complicated, values for all elements likely to be parameterized for semiempirical methods are presented in the Table. From this table, the values of some coefficients are readily derived. Thus for the s-d coulomb integral, Gsd, the coefficient is simply the number of s electrons times the number of d electrons. One s-d exchange integral, Hsd, exists for each electron in the s shell for which there is an electron of the same spin in the d shell. For elements with two s electrons, this is simply the number of d electrons, for elements with one s electron, the Aufbau principle indicates that the d shell with higher occupancy has the same spin as that of the s electron. Finally, the coefficients for the simple d-d repulsion integral, Gdd0Note also that there are no elements with both p and d valence electrons, therefore terms of the type Gpd are not necessary.


Table: Two Electron Energy Contributions to EISOL for Atoms in their Ground States
Element

 

State

 

Gss

Gsp

Hsp

Gpp

Gp2

Gsd

Hsd

Gdd0

Gdd2

Gdd4

 

 

Config.

 

Mult.:

1

1

-1

-1/2

1/2

1

-1/5

1

-1/49

-1/49

1

H

1s1

2S

 

 

 

 

 

 

 

 

 

 

 

2

He

1s2

1S

 

1

 

 

 

 

 

 

 

 

 

3

Li

2s1

2S

 

 

 

 

 

 

 

 

 

 

 

4

Be

2s2

1S

 

1

 

 

 

 

 

 

 

 

 

5

B

2s22p1

2P

 

1

2

1

 

 

 

 

 

 

 

6

C

2s22p2

3P

 

1

4

2

1

3

 

 

 

 

 

7

N

2s22p3

4S

 

1

6

3

3

9

 

 

 

 

 

8

O

2s22p4

3P

 

1

8

4

1

13

 

 

 

 

 

9

F

2s22p5

2P

 

1

10

5

 

20

 

 

 

 

 

10

Ne

2s22p6

1S

 

1

12

6

 

30

 

 

 

 

 

11

Na

3s1

2S

 

 

 

 

 

 

 

 

 

 

 

12

Mg

3s2

1S

 

1

 

 

 

 

 

 

 

 

 

13

Al

3s23p1

2P

 

1

2

1

 

 

 

 

 

 

 

14

Si

3s23p2

3P

 

1

4

2

1

3

 

 

 

 

 

15

P

3s23p3

4S

 

1

6

3

3

9

 

 

 

 

 

16

S

3s23p4

3P

 

1

8

4

1

13

 

 

 

 

 

17

Cl

3s23p5

2P

 

1

10

5

 

20

 

 

 

 

 

18

Ar

3s23p6

1S

 

1

12

6

 

30

 

 

 

 

 

19

K

4s1

2S

 

 

 

 

 

 

 

 

 

 

 

20

Ca

4s2

1S

 

1

 

 

 

 

 

 

 

 

 

21

Sc

4s23d1

2D

 

1

 

 

 

 

2

1

 

 

 

22

Ti

4s23d2

3F

 

1

 

 

 

 

4

2

1

8

1

23

V

4s23d3

4F

 

1

 

 

 

 

6

3

3

15

8

24

Cr

4s13d5

7S

 

 

 

 

 

 

5

5

10

35

35

25

Mn

4s23d5

6S

 

1

 

 

 

 

10

5

10

35

35

26

Fe

4s23d6

5D

 

1

 

 

 

 

12

6

15

35

35

27

Co

4s23d7

4F

 

1

 

 

 

 

14

7

21

43

36

28

Ni

4s23d8

3F

 

1

 

 

 

 

16

8

28

50

43

29

Cu

4s13d10

2S

 

 

 

 

 

 

10

5

45

70

70

30

Zn

4s2

1S

 

1

 

 

 

 

 

 

 

 

 

31

Ga

4s24p1

2P

 

1

2

1

 

 

 

 

 

 

 

32

Ge

4s24p2

3P

 

1

4

2

1

3

 

 

 

 

 

33

As

4s24p3

4S

 

1

6

3

3

9

 

 

 

 

 

34

Se

4s24p4

3P

 

1

8

4

1

13

 

 

 

 

 

35

Br

4s24p5

2P

 

1

10

5

 

20

 

 

 

 

 

36

Kr

4s24p6

1S

 

1

12

6

 

30

 

 

 

 

 

37

Rb

5s1

2S

 

 

 

 

 

 

 

 

 

 

 

38

Sr

5s2

1S

 

1

 

 

 

 

 

 

 

 

 

39

Y

5s24d1

2D

 

1

 

 

 

 

2

1

 

 

 

40

Zr

5s24d2

3F

 

1

 

 

 

 

4

2

1

8

1

41

Nb

5s14d4

6D

 

 

 

 

 

 

4

4

6

21

21

42

Mo

5s14d5

7S

 

 

 

 

 

 

5

5

10

35

35

43

Tc

5s24d5

6S

 

1

 

 

 

 

10

5

10

35

35

44

Ru

5s14d7

5F

 

 

 

 

 

 

7

5

21

43

36

45

Rh

5s14d8

4F

 

 

 

 

 

 

8

5

28

50

43

46

Pd

5s04d10

1S

 

 

 

 

 

 

 

 

45

70

70

47

Ag

5s14d10

2S

 

 

 

 

 

 

10

5

45

70

70

48

Cd

5s2

1S

 

1

 

 

 

 

 

 

 

 

 

49

In

5s25p1

2P

 

1

2

1

 

 

 

 

 

 

 

50

Sn

5s25p2

3P

 

1

4

2

1

3

 

 

 

 

 

51

Sb

5s25p3

4S

 

1

6

3

3

9

 

 

 

 

 

52

Te

5s25p4

3P

 

1

8

4

1

13

 

 

 

 

 

53

I

5s25p5

2P

 

1

10

5

 

20

 

 

 

 

 

54

Xe

5s25p6

1S

 

1

12

6

 

30

 

 

 

 

 

55

Cs

6s1

2S

 

 

 

 

 

 

 

 

 

 

 

56

Ba

6s2

1S

 

1

 

 

 

 

 

 

 

 

 

57

La

6s25d1*

2D*

 

1

 

 

 

 

2

1

 

 

 

72

Hf

6s25d2

3F

 

1

 

 

 

 

4

2

1

8

1

73

Ta

6s25d3

4F

 

1

 

 

 

 

6

3

3

15

8

74

W

6s25d4

5D**

 

1

 

 

 

 

8

4

6

21

21

74 W 6s15d5

7S**

 

 

 

 

 

 

5

5

10

35

35

75

Re

6s25d5

6S

 

1

 

 

 

 

10

5

10

35

35

76

Os

6s25d6

5D

 

1

 

 

 

 

12

6

15

35

35

77

Ir

6s25d7

4F

 

1

 

 

 

 

14

7

21

43

36

78

Pt

6s15d9

3D

 

 

 

 

 

 

9

5

36

56

56

79

Au

6s15d10

2S

 

 

 

 

 

 

10

5

45

70

70

80

Hg

6s2

1S

 

1

 

 

 

 

 

 

 

 

 

81

Tl

6s26p1

2P

 

1

2

1

 

 

 

 

 

 

 

82

Pb

6s26p2

3P

 

1

4

2

1

3

 

 

 

 

 

83

Bi

6s26p3

4S

 

1

6

3

3

9

 

 

 

 

 

84

Po

6s26p4

3P

 

1

8

4

1

13

 

 

 

 

 

85

At

6s25p5

2P

 

1

10

5

 

20

 

 

 

 

 

* The correct ground state of lanthanum is 6s24f1, 2F(u), but for the methods in MOPAC, 
  methods that do not use "f" orbitals,  the low-lying excited state 6s25d1, 2D(g), is used instead.

** The lowest state of tungsten is 5D0, but the lowest state ignoring spin-orbit coupling is 7S3
MOPAC does not use relativistic terms, so spin-orbit splitting is not considered. 
Because of this, the ground state used in MOPAC for W is 7S.