Feature&Fix: Correct implementation of length-gauge electric fields and ionic forces in RT-TDDFT​ (Useful information for fixed-bug version of rt-TDDFT)

[AsTonyshment]

Description

This PR addresses a critical implementation error in length-gauge RT-TDDFT.

1. Critical Bug Fix

• The original implementation incorrectly assumed cubic lattice symmetry (orthogonal axes with a=b=c). Calculations were accidentally correct only for cubic systems due to hidden simplifications in the coordinate transformations. Now it can properly handle arbitrary lattice vectors.
• Correct calculation of ionic forces under multi-directional electric fields and arbitrary field combinations (e.g., simultaneous x+y+z directions with mixed Gaussian/Heaviside pulses).

2. Validation

• Verified against non-cubic lattices and multi-directional field. A system of single He atom is used (since it is electrically neutral, the resultant force it experiences in an electric field should be zero). Results (fcs is the total force, and fefield_tddft is the ionic electric field force, the unit is eV/Å):
  

3. Completely rewritten RT-TDDFT documentation

[ESROAMER]

The validation looks good. The result is proved to be internally consistent in the length gauge. But I still want to confirm the details about the direction of the electric field. Generally, we work with two types of coordinates: fractional(direct) coordinates and Cartesian coordinates, the latter always being orthogonal. The conversion between the two coordinate systems is given by:
$x_{c}=Ax_{d}$
where A denotes the matrix of lattice vectors. I noticed your changes at line 195 in H_TDDFT_pw.cpp. I suppose it assumes the electric field direction in INPUT parameters is in the direct coordinates, and here it's being converted to Cartesian coordinates? The vector potentials in the velocity gauge and the hybrid gauge don't involve additional directional transformations and align with the direction of $E(t)$. In the velocity gauge, the direction is actually determined by the interface of gradient, while in the hybrid gauge it's actually determined by the interface of r matrix. Both gauges treat the input electric field direction as being along the Cartesian coordinate system by default.(I'm not entirely certain.) Therefore, there might be inconsistencies in the default electric field directions across different gauges here.
I suggest we might need cross-gauge testing for verification, though these tests could certainly be conducted after this PR gets merged.

[AsTonyshment]

@ESROAMER Yes, this PR changed the direction of electric field in length gauge to Cartesian coordinates, which should be consistent with velocity gauge and hybrid gauge. Line 195 basically means projecting to the Cartesian x-axis:
$$\vec{E}_x\cdot \vec{r}=E\vec{e}_x\cdot (i \vec{a}+j \vec{b}+k \vec{c})=E(i a_x+j b_x+ k c_x)$$
where $\vec{a},\vec{b},\vec{c}$ are lattice vectors, and $(i,j,k)$ are lattice fractional coordinate of a space point $\vec{r}$. You can also see this in the change of the documentation of this PR, in which I made furthermore clarification:

But I think your suggestion is right, I will conduct a cross-gauge testing to make sure everything is right in the near future.

[ESROAMER]

I see, thanks for your clarification. I'll do some tests as well.