*So you take a picture of something you see*

In the future where will I be?

You can climb a ladder up to the sun

Or write a song nobody has sung

Or do something that's never been done

Or do something that's never been done

--Coldplay, "Talk"

In the future where will I be?

You can climb a ladder up to the sun

Or write a song nobody has sung

Or do something that's never been done

Or do something that's never been done

--Coldplay, "Talk"

## My interest

- Scientifically, it is amazing that many (electronic,
mechanical, optical) properties calculated from such
**first principle theories**(i.e. the only input is atomic specicies and atomic positions. There is in principle no dependence on empirical or fitting parameters) can be directly compared to experimental results. - Mathematically, the theories are relatively well defined. They also provide mathematical and computational challenges not commonly seen in other branches of applied mathematics. The complexity of electronic structure theories can range from high (like density functional theory) to very high (like configuration interaction), and hence numerical methods with reduced complexity are urgently needed in order to simulate ever larger relastic systems.
- This field had been active in physics and chemistry before computers even existed, and yet the field is still at its very early stage from the perspective of applied mathematics. In my view, "language barriers" are sometimes the biggest obstacle preventing mathematicians from understanding the problem. There is great potential for numerical analysis and numerical linear algebra tools to deeply impact this field.

## Introductory material

**it does not assume that you already know quantum mechanics**(we start from scratch and then move very quickly).

The video of the lectures is available on the MSRI website (2.5 hours each day for 10 days).

Additional reading, matlab codes and projects can be found on the LBNL site.

## Recent Research Directions

**Please see the "Publication" and "Presentation" tabs for more details.**

### Adaptive compression schemes

The **adaptively compressed exchange operator (ACE)**
formulation reduces the computational cost associated with the Fock exchange
operator in Hartree-Fock and hybrid functional Kohn-Sham calculations. The ACE
formulation does not depend on the size of the band gap, and thus can be
applied to insulating, semiconducting as well as metallic systems, and can
reduce the cost of hybrid functional calculations by 5-10 fold without loss of
accuracy. In an iterative framework for solving Hartree-Fock-like systems, the
ACE formulation only requires moderate modification of the code.

The ACE formulation for Fock exchange calculation is adopted in a
number of community electronic structure software packages such as Quantum
ESPRESSO and PETot.

The concept of adaptive compression also finds its use in compressing the polarizability operator
in phonon calculations. The **adaptively compressed polarizability (ACP)** formulation reduces the
cost of phonon calculations from \(\mathcal{O}(N^4)\) to \(\mathcal{O}(N^3)\).

### QRCP based compression schemes

QR factorization with column pivoting (QRCP) is an overlooked and
yet powerful technique for electronic structure calculation.
We have developed the
**selected columns of the density matrix (SCDM)** formulation
for finding localized
representation of the subspace spanned by occupied Kohn-Sham
orbitals for insulating systems using QRCP. SCDM explicitly uses
the fact that density matrices associated with insulating
systems decay exponentially along the off-diagonal direction
in the real space representation, and avoid the usage of an
optimization procedure. SCDM is simple to code (two lines of
MATLAB code for demonstration purpose!), and is efficient, robust and
massively parallelizable.

The interface of SCDM with Wannier90 package is under progress.

### Multiscale methods (embedding)

For mesoscopic systems, the global computational domain is very large.
Due to the limit of computational power, only certain degrees of freedoms
can be treated accurately with high fidelity quantum
mechanical theories. The rest of the domain
must be modeled using some lower fidelity theories. In physics, this is referred to as the ``embedding'' problem.
We developed the **PEXSI-\(\Sigma\) method** (see introduction
to the PEXSI method below) for coupling quantum mechanical calculations
to classical calculations. The PEXSI-\(\Sigma\) method is a discrete analog of the
Dirichlet-to-Neumann mapping (DtN) method, where the DtN operator is constructed using
a reference Green's function from a crystal configuration.

### Spectral density estimation

Methods for estimating the **spectral density (a.k.a. density of states)** only use
matrix-vector multiplication, and can be powerful techniques when
diagonalization of the full matrix is prohibitively expensive.
They can be useful for computing excited state properties
without fully diagonalizing the matrix, or even without having the matrix explicitly constructed.
This is the case e.g. when the
absorption spectrum is to be computed in
linear response time-dependent
density functional theory (TDDFT), or when the Bethe-Salpeter equation (BSE)
is to be solved.

The convergence of randomized methods can also be accelerated using
the **spectrum sweeping**
method, which sweep
through the entire spectrum by building such low rank
decomposition at different parts of the spectrum.

### Fast Green's function methods

The **pole expansion and selected inversion (PEXSI)**
method is an alternative
way for solving KSDFT without diagonalizing the Hamiltonian
operator.
The pole expansion converts the problem of evaluating
the electron density to the problem of
evaluating the selected elements of Green's functions
(i.e. inverse matrices). The PEXSI method
can be used for both metallic and insulating systems. PEXSI
scales as \(O(N)\) for quasi-1D systems,
\(O(N^{1.5})\) for quasi-2D systems, \(O(N^2)\) for 3D
bulk systems, and therefore overcomes the
\(\mathcal{O}(N^3)\) barrier in standard Kohn-Sham dFT calculations.

The PEXSI software package has
been integrated into community electronic structure packages such as SIESTA, CP2K, DGDFT,
BigDFT, and Atomistix ToolKit, and can harness more than 100,000 cores on the
NERSC system at DOE for systems over 20,000 atoms. It has been used for studying large scale
2D materials such as graphene nanoflake, phospherene nanoflake,
and for predicting phospherene nanoflake heterojunction as highly
efficient solar cells materials.

### Adaptive basis set

The **Discontinuous Galerkin
Density Functional Theory (DGDFT) **
uses adaptive local basis functions to discretize the Kohn-Sham
Hamiltonian. These basis functions are localized in the real space, and are
discontinuous in the global
domain. DGDFT can yield both systematically improvable results
while keeping a relatively small basis set, with rigorous a
posteriori error analysis.

The DGDFT package can scale beyond 100,000 processors and solves
electronic structure problem with more than 20,000 atoms. The
recent wall clock time per SCF for a graphene system with more
than 10,000 atoms is around one minute.