What is the physical meaning of the term "non-local interaction"?

Question

This term pops up through out Hartree-Fock and DFT. Does this term have any physical significance to actual(real) system or it is a mathematical by-product?

Answer 1

The non-local operator in Hartree Fock is the Exchange operator. Locality is a mathematical property. Look at the Coulomb operator first which acts like this: $$ \hat v_H\;\varphi_s(r) = \sum_i^{N_\textrm{occ}}\int dr' \frac{\varphi_i(r') \varphi_i(r')}{|r-r'|} \varphi_s(r).\tag{1} $$

It's value depends for the function (which I call here $\varphi_s$) on which it acts only on the value of the function at $r$.

Now look at the Exchange Operator which acts like this:

$$ \hat v_x\;\varphi_s(r) = \sum_i^{N_\textrm{occ}}\int dr' \frac{\varphi_i(r') \varphi_s(r')}{|r-r'|} \varphi_i(r).\tag{2} $$

As you can see $\varphi_s$ is now inside the integral, in order to evaluate the "effect" of the operator you will need knowledge the function $\varphi_s$ at all positions of space. Hence the designation non-local.

The physical meaning of it is that it reflects the permutation symmetry of the fermionic system. This effect is sometimes called Fermi-correlation. Just follow the derivation of Hartree Fock and it will become obvious. A book which contains this derivation is "Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory (Dover Books on Chemistry)"

$\varphi$ is a Fock orbital. The index $i$ refers to occupied orbitals. The index $s$ refers to all orbitals. $r$ is the coordinate of a particle in space while $r'$ is the coordinate of another particle in space. $v_H$ is the Coulomb Potential, $\hat v_x$ is the exchange potential. Often in the context of Hartree Fock $\hat v_H$ is denoted as $\hat J$ and $\hat v_x$ is denoted as $\hat K$.

Answer 2

It's all hollow words if no proper definition is given. The definition I know is the following: For a one-body operator $A$, define $$A(x,x^\prime):=\langle x|A|x^\prime\rangle \tag{1} \quad . $$

We call $A$ local if the following holds:

$$ A(x,x^\prime)= \delta(x-x^\prime)\, A(x),\tag{2} $$

and otherwise it's non-local. Here we employed a usual abuse of notation in equation $(2)$. An example for a local operator would be a local external potential, e.g. the external potential for an electron interacting with some nuclei in the Born-Oppenheimer approximation.

Whether or not an external potential is local becomes important in e.g. the Hohenberg-Kohn theorem.

Reference: Density-Functional Theory of Atoms and Molecules. Parr and Yang. 1989, section 2.1, p.23, equation (2.13) and (2.14) as well as the corresponding discussion.