1.
Explain the following.
a.
What is the electron density?
Electron density is the measure of the probability of an electron being present at a
specific location.
b.
What is a functional?
A functional is a function of a “definite” function
. Mathematical operator where
the input is a function and the output is a number.
c.
What is the exchange-correlation functional?
The exchange-correlation functional is a correction term. We made assumptions
about the non-interacting electrons because this assumption is wrong. This
assumption allowed us to get energies in the earlier states of the equation, but it
gives us an error. The exchange correlation function is added to help correct that
error. This functional is unknown and the best we can do is make approximations.

2.
Evaluate the following functionals.
a.
g(x)=x
4
i.
∫ 𝑥
4
?𝑥 =
𝑥
5
5
=
1
5
1
0
ii.
𝑑
2
𝑑𝑥
2
𝑥
4
= 12𝑥
2
= 12(1) = 12
b.
g(x)=sin(-x)
i.
∫
sin(−𝑥) ?𝑥 = cos(𝑥) = cos(1) − cos(0) = −0.459
1
0
ii.
𝑑
2
𝑑𝑥
2
sin(−𝑥) = sin(𝑥) = sin(1) = 0.8414
c.
g(x)=exp(-a*x
2
)
i.
∫ ?
−𝑎∗𝑥
2
?𝑥 =
√𝜋
2∗√𝑎
erf(𝑎 ∗ √𝑥
) =
1
0
√𝜋
2∗√𝑎
erf(𝑎 ∗ √1
) =
√𝜋
2∗√𝑎
erf(𝑎)
ii.
𝑑
2
𝑑𝑥
2
?
−𝑎∗𝑥
2
= −2a(−2𝑎?
−𝑎𝑥
2
𝑥
2
+ ?
−𝑎𝑥
2
) = −2a(−2𝑎?
−𝑎1
2
1
2
+
?
−𝑎1
2
) = 2a(−2𝑎?
−𝑎
+ ?
−𝑎
)

3.
Explain
the
difference
between
GGA,
meta-GGA,
and
Hybrid
exchange-correlation
functionals. Give examples of each.
GGA relies on electron density and the curvature of electron density. How the electron
density changing and what is at a given point. One example is BLYP. The meta-GGA

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