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