Analytical Jacobi rotations for AP1roG

We have an analytical expression for the AP1roG-energy: EAP1roG=NPj(2hjj+NPk(2gkkjjgkjjk)+Kb=NP+1gjbjbGbj). Using Jacobi rotations, we can calculate an analytical expression for the rotated energy by means of a Jacobi rotation. The rotated energy is given by EAP1roG, rotated(p,q,θ)=EAP1roG+Erotation(p,q,θ), in which the expressions for Erotation(p,q,θ) will be listed in this section. For convenience, we will drop the p and q arguments.

Let us first state that our Jacobi rotations are only defined for 1q<pK. Furthermore, since the summations in the expression of the AP1roG-energy are not over all basis functions K, we have to distinguish three disjoint cases for the rotation energy Erotation(θ):

  1. qNPpNP, which are occupied-occupied rotations. In this case, the rotated energy is E1(θ)=A1+B1cos(2θ)+C1sin(2θ), in which A1=12Kb=NP+1(gbpbpgbqbq)(GbpGbq),B1=12Kb=NP+1(gbpbpgbqbq)(GbpGbq)=A1, and C1=Kb=NP+1gbpbq(GbqGbp).

The minimum of E1(θ) can be found at θmin,1=12arctan(B1B21+C21,C1B21+C21).

  1. p>NPqNP, being occupied-virtual rotations. In this case, the rotated energy is a little more complicated, being E2(θ)=A2+B2cos(2θ)+C2sin(2θ)+D2cos(4θ)+E2sin(4θ), in which A2=hpphqq+38(gpppp+gqqqq)(1Gpq)14gppqq(7+Gpq)+12gpqpq(3+Gpq)+NPj(2(gjjppgjjqq)12(gjpjpgjqjq)(2+Gpj))+12Kb=NP(gbpbpgbqbq)Gbq,B2=hqqhpp+2gppqq+12(gpppp+gqqqq)(Gpq1)gpqpq(1+Gpq)+NPj(2(gjjqqgjjpp)+12(gjpjpgjqjq)(2+Gpj))+12Kb=NP(gbqbqgbpbp)Gbq,C2=2hpq+(gpppqgpqqq)(1Gpq)+NPj(4gjjpqgjpjq(2+Gpj))+Kb=NPgbpbqGbq,D2=18(gpppp+gqqqq2(gppqq+2gpqpq))(1Gpq), and E2=12(gpppqgpqqq)(Gpq1).

In principle, we could write down the minimum of E2(θ) analytically, but Mathematica lists a tremendously long expression, so I think we are better off using a Newton-step based approach, since we have easy access to gradient and Hessian. Its first derivative is dE(θ)dθ=2B2sin(2θ)+2C2cos(2θ)4D2sin(4θ)+4E2cos(4θ) and the second derivative is d2E(θ)dθ2=4B2cos(2θ)4C2sin(2θ)16D2cos(4θ)16E2sin(4θ).

Since the expression of the energy after a Jacobi rotation (???) has a period of π, we can confine our search for an optimal θ to the interval [π2,π2[. We can also note that the extremal points of this interval, by the definition of our Jacobi rotations, correspond to the case in which the two orbitals are merely switched (and one of them gets a sign change).

  1. p>NPq>NP, of course taking into account equation (???), these are virtual-virtual rotations. In this case, the rotated energy is again simple: E3(θ)=A3+B3cos(2θ)+C3sin(2θ), where A3=12NPj(gjpjpgjqjq)(GpjGqj),B3=12NPj(gjpjpgjqjq)(GpjGqj)=A3, and C3=NPjgjpjq(GqjGpj).

The minimum of E3(θ) is very analogous to the first case: it can be found at θmin,3=12arctan(B3B23+C23,C3B23+C23). Note that the case pNPq>NP is forbidden by equation (???).