- Source: 1s Slater-type function
A normalized 1s Slater-type function is a function which is used in the descriptions of atoms and in a broader way in the description of atoms in molecules. It is particularly important as the accurate quantum theory description of the smallest free atom, hydrogen. It has the form
ψ
1
s
(
ζ
,
r
−
R
)
=
(
ζ
3
π
)
1
2
e
−
ζ
|
r
−
R
|
.
{\displaystyle \psi _{1s}(\zeta ,\mathbf {r-R} )=\left({\frac {\zeta ^{3}}{\pi }}\right)^{1 \over 2}\,e^{-\zeta |\mathbf {r-R} |}.}
It is a particular case of a Slater-type orbital (STO) in which the principal quantum number n is 1. The parameter
ζ
{\displaystyle \zeta }
is called the Slater orbital exponent. Related sets of functions can be used to construct STO-nG basis sets which are used in quantum chemistry.
Applications for hydrogen-like atomic systems
A hydrogen-like atom or a hydrogenic atom is an atom with one electron. Except for the hydrogen atom itself (which is neutral) these atoms carry positive charge
e
(
Z
−
1
)
{\displaystyle e(\mathbf {Z} -1)}
, where
Z
{\displaystyle \mathbf {Z} }
is the atomic number of the atom. Because hydrogen-like atoms are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be exactly solved in analytic form. The solutions are one-electron functions and are referred to as hydrogen-like atomic orbitals.
The electronic Hamiltonian (in atomic units) of a Hydrogenic system is given by
H
^
e
=
−
∇
2
2
−
Z
r
{\displaystyle \mathbf {\hat {H}} _{e}=-{\frac {\nabla ^{2}}{2}}-{\frac {\mathbf {Z} }{r}}}
, where
Z
{\displaystyle \mathbf {Z} }
is the nuclear charge of the hydrogenic atomic system. The 1s electron of a hydrogenic systems can be accurately described by the corresponding Slater orbital:
ψ
1
s
=
(
ζ
3
π
)
0.50
e
−
ζ
r
{\displaystyle \mathbf {\psi } _{1s}=\left({\frac {\zeta ^{3}}{\pi }}\right)^{0.50}e^{-\zeta r}}
, where
ζ
{\displaystyle \mathbf {\zeta } }
is the Slater exponent. This state, the ground state, is the only state that can be described by a Slater orbital. Slater orbitals have no radial nodes, while the excited states of the hydrogen atom have radial nodes.
= Exact energy of a hydrogen-like atom
=The energy of a hydrogenic system can be exactly calculated analytically as follows :
E
1
s
=
⟨
ψ
1
s
|
H
^
e
|
ψ
1
s
⟩
⟨
ψ
1
s
|
ψ
1
s
⟩
{\displaystyle \mathbf {E} _{1s}={\frac {\langle \psi _{1s}|\mathbf {\hat {H}} _{e}|\psi _{1s}\rangle }{\langle \psi _{1s}|\psi _{1s}\rangle }}}
, where
⟨
ψ
1
s
|
ψ
1
s
⟩
=
1
{\displaystyle \mathbf {\langle \psi _{1s}|\psi _{1s}\rangle } =1}
E
1
s
=
⟨
ψ
1
s
|
−
∇
2
2
−
Z
r
|
ψ
1
s
⟩
{\displaystyle \mathbf {E} _{1s}=\langle \psi _{1s}|\mathbf {-} {\frac {\nabla ^{2}}{2}}-{\frac {\mathbf {Z} }{r}}|\psi _{1s}\rangle }
E
1
s
=
⟨
ψ
1
s
|
−
∇
2
2
|
ψ
1
s
⟩
+
⟨
ψ
1
s
|
−
Z
r
|
ψ
1
s
⟩
{\displaystyle \mathbf {E} _{1s}=\langle \psi _{1s}|\mathbf {-} {\frac {\nabla ^{2}}{2}}|\psi _{1s}\rangle +\langle \psi _{1s}|-{\frac {\mathbf {Z} }{r}}|\psi _{1s}\rangle }
E
1
s
=
⟨
ψ
1
s
|
−
1
2
r
2
∂
∂
r
(
r
2
∂
∂
r
)
|
ψ
1
s
⟩
+
⟨
ψ
1
s
|
−
Z
r
|
ψ
1
s
⟩
{\displaystyle \mathbf {E} _{1s}=\langle \psi _{1s}|\mathbf {-} {\frac {1}{2r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial }{\partial r}}\right)|\psi _{1s}\rangle +\langle \psi _{1s}|-{\frac {\mathbf {Z} }{r}}|\psi _{1s}\rangle }
. Using the expression for Slater orbital,
ψ
1
s
=
(
ζ
3
π
)
0.50
e
−
ζ
r
{\displaystyle \mathbf {\psi } _{1s}=\left({\frac {\zeta ^{3}}{\pi }}\right)^{0.50}e^{-\zeta r}}
the integrals can be exactly solved. Thus,
E
1
s
=
⟨
(
ζ
3
π
)
0.50
e
−
ζ
r
|
−
(
ζ
3
π
)
0.50
e
−
ζ
r
[
−
2
r
ζ
+
r
2
ζ
2
2
r
2
]
⟩
+
⟨
ψ
1
s
|
−
Z
r
|
ψ
1
s
⟩
{\displaystyle \mathbf {E} _{1s}=\left\langle \left({\frac {\zeta ^{3}}{\pi }}\right)^{0.50}e^{-\zeta r}\right|\left.-\left({\frac {\zeta ^{3}}{\pi }}\right)^{0.50}e^{-\zeta r}\left[{\frac {-2r\zeta +r^{2}\zeta ^{2}}{2r^{2}}}\right]\right\rangle +\langle \psi _{1s}|-{\frac {\mathbf {Z} }{r}}|\psi _{1s}\rangle }
E
1
s
=
ζ
2
2
−
ζ
Z
.
{\displaystyle \mathbf {E} _{1s}={\frac {\zeta ^{2}}{2}}-\zeta \mathbf {Z} .}
The optimum value for
ζ
{\displaystyle \mathbf {\zeta } }
is obtained by equating the differential of the energy with respect to
ζ
{\displaystyle \mathbf {\zeta } }
as zero.
d
E
1
s
d
ζ
=
ζ
−
Z
=
0
{\displaystyle {\frac {d\mathbf {E} _{1s}}{d\zeta }}=\zeta -\mathbf {Z} =0}
. Thus
ζ
=
Z
.
{\displaystyle \mathbf {\zeta } =\mathbf {Z} .}
= Non-relativistic energy
=The following energy values are thus calculated by using the expressions for energy and for the Slater exponent.
Hydrogen : H
Z
=
1
{\displaystyle \mathbf {Z} =1}
and
ζ
=
1
{\displaystyle \mathbf {\zeta } =1}
E
1
s
=
{\displaystyle \mathbf {E} _{1s}=}
−0.5 Eh
E
1
s
=
{\displaystyle \mathbf {E} _{1s}=}
−13.60569850 eV
E
1
s
=
{\displaystyle \mathbf {E} _{1s}=}
−313.75450000 kcal/mol
Gold : Au(78+)
Z
=
79
{\displaystyle \mathbf {Z} =79}
and
ζ
=
79
{\displaystyle \mathbf {\zeta } =79}
E
1
s
=
{\displaystyle \mathbf {E} _{1s}=}
−3120.5 Eh
E
1
s
=
{\displaystyle \mathbf {E} _{1s}=}
−84913.16433850 eV
E
1
s
=
{\displaystyle \mathbf {E} _{1s}=}
−1958141.8345 kcal/mol.
= Relativistic energy of Hydrogenic atomic systems
=Hydrogenic atomic systems are suitable models to demonstrate the relativistic effects in atomic systems in a simple way. The energy expectation value can calculated by using the Slater orbitals with or without considering the relativistic correction for the Slater exponent
ζ
{\displaystyle \mathbf {\zeta } }
. The relativistically corrected Slater exponent
ζ
r
e
l
{\displaystyle \mathbf {\zeta } _{rel}}
is given as
ζ
r
e
l
=
Z
1
−
Z
2
/
c
2
{\displaystyle \mathbf {\zeta } _{rel}={\frac {\mathbf {Z} }{\sqrt {1-\mathbf {Z} ^{2}/c^{2}}}}}
.
The relativistic energy of an electron in 1s orbital of a hydrogenic atomic systems is obtained by solving the Dirac equation.
E
1
s
r
e
l
=
−
(
c
2
+
Z
ζ
)
+
c
4
+
Z
2
ζ
2
{\displaystyle \mathbf {E} _{1s}^{rel}=-(c^{2}+\mathbf {Z} \zeta )+{\sqrt {c^{4}+\mathbf {Z} ^{2}\zeta ^{2}}}}
.
Following table illustrates the relativistic corrections in energy and it can be seen how the relativistic correction scales with the atomic number of the system.
References
Kata Kunci Pencarian:
- 1s Slater-type function
- Slater's rules
- Gaussian orbital
- Basis set (chemistry)
- Effective nuclear charge
- STO-nG basis sets
- Atomic orbital
- Index of physics articles (0–9)
- Electron configuration
- Ionization energy
