Mathcad - qlab-scf-gauss.mcd - Users.csbsju. edu

SCF Calculation for Two Electron Atoms and Ions             Using a Gaussian Wave Function 3 4

⎛ 2⋅ β ⎞ 2⎞ ⎛ Ψ ( r , β ) := ⎜ ⎟ ⋅ exp ⎝−β ⋅ r ⎠ ⎝ π ⎠

Gaussian Trial Wave Function:

∞

Calculate kinetic energy:

⌠ ⎮ ⎤ ⎡ 2 2 ⎢ 1 d ⎥ ⎮ Ψ (r , β)⋅ − ⎢ 2⋅ r ⋅ 2 ( r⋅ Ψ ( r , β ) )⎥ ⋅ 4⋅ π ⋅ r dr ⎮ dr ⎣ ⎦ ⎮ ⌡0

assume , β > 0 3 → ⋅β simplify 2

Calculate electron‐nucleus potential energy: 1 ∞

⌠ −Z 2 ⎮ Ψ (r , β)⋅ ⋅ Ψ ( r , β ) ⋅ 4⋅ π ⋅ r dr ⎮ r ⌡0

1

2

assume , β > 0 2 2 → ( − 2) ⋅ ⋅(β⋅π) ⋅Z simplify π

Calculation of electron‐electron potential energy: a. Calculate the electric potential of one of the electrons in the presence of the other:

1 ⌠ ⎮ ⋅⎮ r ⌡

r

0

∞

⌠ ⎮ 2 2 Ψ ( x , β ) ⋅ 4⋅ π ⋅ x dx + ⎮ ⎮ ⌡r

⎛⎜ 1 1 ⎞⎟ 2 2 assume , β > 0 erf ⎜ r ⋅ 2 2 ⋅ β 2 ⎟ Ψ ( x , β ) ⋅ 4⋅ π ⋅ x ⎝ ⎠ → dx x

simplify

r

b. Calculate the electron‐electron potential energy using result of part a: ∞

⌠ ⎮ ⎛ ⎛⎜ 1 1 ⎞⎟ ⎞ ⎜ ⎟ ⎮ 2 2⎟ ⎜ ⎮ 2 ⎜ erf ⎝ r ⋅ 2 ⋅ β ⎠ ⎟ 2 ⎮ Ψ (r , β) ⋅⎜ ⎟ ⋅ 4 ⋅ π ⋅ r dr r ⎝ ⎠ ⎮ ⌡0

1

assume , β > 0 2 2 → ⋅(β⋅π) π simplify

SCF Calculation Z := 2

1. Supply nuclear charge and an input value for β:

β := 0.7670

α := Z

2. Define orbital energies of the electrons in terms of the variational parameters:

ε 1sα ( α , β ) :=

Orbital energy of the α electron:

ε 1sβ ( α , β ) :=

Orbital energy of the β electron:

3⋅ α 2 3⋅ β 2

− Z⋅

− Z⋅

8⋅ α π 8⋅ β π

+

+

8⋅ α ⋅ β π(α + β) 8⋅ α ⋅ β π(α + β)

3. Minimize orbital energies with respect to α and β:

Given

Given

d dα d dβ

ε 1sα ( α , β ) = 0

α := Find ( α )

α = 0.7670

ε 1sα ( α , β ) = −0.6564

ε 1sβ ( α , β ) = 0

β := Find ( β )

β = 0.7670

ε 1sβ ( α , β ) = −0.6564

4. Calculate the energy of the atom:

E atom :=

3⋅ α 2

+

3⋅ β 2

− Z⋅

8⋅ α π

− Z⋅

8⋅ β π

+

8⋅ α ⋅ β

E atom = −2.3010

π(α + β)

5. Record results of the SCF cycle and return to step 1 with the new and improved input value for β. 6. Continue until self‐consistency is achieved. 7. Verify the results shown below for He.  Repeat for Li+, Be2+  and B3+.

⎛β ( input) ⎜ ⎜ 2.0000 ⎜ ⎜ 0.9303 ⎜ 0.8023 ⎜ ⎜ 0.7749 ⎜ ⎜ 0.7688 ⎜ 0.7674 ⎜ ⎜ 0.7671 ⎜ ⎝ 0.7670

α

ε 1sα

β

ε 1sβ

0.4514 −0.4988 0.9303 −0.8031 0.6946 −0.6117 0.8023 −0.6816 0.7504 −0.6454 0.7749 −0.6618 0.7633 −0.6539 0.7688 −0.6576 0.7661 −0.6558 0.7674 −0.6567 0.7668 −0.6563 0.7671 −0.6564 0.7669 −0.6564 0.7670 −0.6564 0.7670 −0.6564 0.7670 −0.6564

⎞ ⎟ −2.2703 ⎟ ⎟ −2.2996 ⎟ −2.3009 ⎟ ⎟ −2.3010 ⎟ ⎟ −2.3010 ⎟ −2.3010 ⎟ ⎟ −2.3010 ⎟ ⎟ −2.3010 ⎠ E atom