Feature: RI-Hartree for LR-TDDFT benchmark with FHI-aims

[maki49]

What's the feature for

Read LRI $C$-tensors and Coulomb matrix $V$ under FHI-aims' ABFs and KS-eigenstates $c$ under FHI-aims' NAOs into ABACUS, then transform the Coulomb matrix directly from ABFs to particle-hole basis by $c_ic_aCVCc_jc_b$ to get the Hartree term in Casida A matrix comparable with FHI-aims, bypassing the difference between the NAO basis set used in the two software.

What bugs have been found and fixed by this feature

• coefficients of the Fock term
• wrong indexes of the transition density matrix in the Fock term
• the default ccp_type that should be consistent with xc_kernel

Some test results

H2	TDRPA@HF		TDHF@HF
aims, RI-V	singlet	triplet	singlet	triplet
eig 1(eV)	25.1312	21.8674	14.3507	10.8087
eig 2(eV)	35.1996	30.1516	22.649	17.1259
eig 3(eV)	48.301	42.6014	32.2035	26.0892
abacus-read-aims, RI-LVL	singlet	triplet	singlet	triplet
eig 1(eV)	25.3704	21.8643	14.4533	10.9472
eig 2(eV)	35.6267	30.1529	22.8164	17.3427
eig 3(eV)	48.6821	42.6009	32.3465	26.2623

H2O	TDRPA@HF		TDHF@HF
abacus, Gint	singlet	triplet	singlet	triplet
eig 1(eV)	19.9876	19.2461	10.2693	9.53565
eig 2(eV)	24.096	23.5622	12.8549	12.3211
eig 3(eV)	30.093	29.4697	18.746	16.8614
abacus-read-abacus, RI-LVL	singlet	triplet	singlet	triplet
eig 1(eV)	19.9916	19.2461	10.2729	9.5357
eig 2(eV)	24.092	23.5622	12.8509	12.3211
eig 3(eV)	30.0933	29.4697	18.7467	16.8614

How to use it

See the addition in the documentation and the new example.

@Fisherd99 FYI