# Projector augmented wave method

Last updated

The projector augmented wave method (PAW) is a technique used in ab initio electronic structure calculations. It is a generalization of the pseudopotential and linear augmented-plane-wave methods, and allows for density functional theory calculations to be performed with greater computational efficiency. [1]

## Contents

Valence wavefunctions tend to have rapid oscillations near ion cores due to the requirement that they be orthogonal to core states; this situation is problematic because it requires many Fourier components (or in the case of grid-based methods, a very fine mesh) to describe the wavefunctions accurately. The PAW approach addresses this issue by transforming these rapidly oscillating wavefunctions into smooth wavefunctions which are more computationally convenient, and provides a way to calculate all-electron properties from these smooth wavefunctions. This approach is somewhat reminiscent of a change from the Schrödinger picture to the Heisenberg picture.

## Transforming the wavefunction

The linear transformation ${\displaystyle {\mathcal {T}}}$ transforms the fictitious pseudo wavefunction ${\displaystyle |{\tilde {\Psi }}\rangle }$ to the all-electron wavefunction ${\displaystyle |\Psi \rangle }$:

${\displaystyle |\Psi \rangle ={\mathcal {T}}|{\tilde {\Psi }}\rangle }$

Note that the "all-electron" wavefunction is a Kohn–Sham single particle wavefunction, and should not be confused with the many-body wavefunction. In order to have ${\displaystyle |{\tilde {\Psi }}\rangle }$ and ${\displaystyle |\Psi \rangle }$ differ only in the regions near the ion cores, we write

${\displaystyle {\mathcal {T}}=1+\sum _{R}{\hat {\mathcal {T}}}_{R}}$,

where ${\displaystyle {\hat {\mathcal {T}}}_{R}}$ is non-zero only within some spherical augmentation region ${\displaystyle \Omega _{R}}$ enclosing atom ${\displaystyle R}$.

Around each atom, it is useful to expand the pseudo wavefunction into pseudo partial waves:

${\displaystyle |{\tilde {\Psi }}\rangle =\sum _{i}|{\tilde {\phi }}_{i}\rangle c_{i}}$ within ${\displaystyle \Omega _{R}}$.

Because the operator ${\displaystyle {\mathcal {T}}}$ is linear, the coefficients ${\displaystyle c_{i}}$ can be written as an inner product with a set of so-called projector functions, ${\displaystyle |p_{i}\rangle }$:

${\displaystyle c_{i}=\langle p_{i}|{\tilde {\Psi }}\rangle }$

where ${\displaystyle \langle p_{i}|{\tilde {\phi }}_{j}\rangle =\delta _{ij}}$. The all-electron partial waves, ${\displaystyle |\phi _{i}\rangle ={\mathcal {T}}|{\tilde {\phi }}_{i}\rangle }$, are typically chosen to be solutions to the Kohn–Sham Schrödinger equation for an isolated atom. The transformation ${\displaystyle {\mathcal {T}}}$ is thus specified by three quantities:

1. a set of all-electron partial waves ${\displaystyle |\phi _{i}\rangle }$
2. a set of pseudo partial waves ${\displaystyle |{\tilde {\phi }}_{i}\rangle }$
3. a set of projector functions ${\displaystyle |p_{i}\rangle }$

and we can explicitly write it down as

${\displaystyle {\mathcal {T}}=1+\sum _{i}\left(|\phi _{i}\rangle -|{\tilde {\phi }}_{i}\rangle \right)\langle p_{i}|}$

Outside the augmentation regions, the pseudo partial waves are equal to the all-electron partial waves. Inside the spheres, they can be any smooth continuation, such as a linear combination of polynomials or Bessel functions.

The PAW method is typically combined with the frozen core approximation, in which the core states are assumed to be unaffected by the ion's environment. There are several online repositories of pre-computed atomic PAW data. [2] [3] [4]

## Transforming operators

The PAW transformation allows all-electron observables to be calculated using the pseudo-wavefunction from a pseudopotential calculation, conveniently avoiding having to ever represent the all-electron wavefunction explicitly in memory. This is particularly important for the calculation of properties such as NMR, [5] which strongly depend on the form of the wavefunction near the nucleus. Starting with the definition of the expectation value of an operator:

${\displaystyle a_{i}=\langle \Psi |{\hat {A}}|\Psi \rangle }$,

where you can substitute in the pseudo wavefunction as you know ${\displaystyle |\Psi \rangle ={\mathcal {T}}|{\tilde {\Psi }}\rangle }$:

${\displaystyle a_{i}=\langle {\tilde {\Psi }}|{\mathcal {T}}^{\dagger }{\hat {A}}{\mathcal {T}}|{\tilde {\Psi }}\rangle }$,

from which you can define the pseudo operator, indicated by a tilde:

${\displaystyle {\tilde {A}}={\mathcal {T}}^{\dagger }{\hat {A}}{\mathcal {T}}}$.

If the operator ${\displaystyle {\hat {A}}}$ is local and well-behaved we can expand this using the definition of ${\displaystyle {\mathcal {T}}}$ to give the PAW operator transform

${\displaystyle {\tilde {A}}={\hat {A}}+\sum _{i,j}|p_{i}\rangle \left(\langle \phi _{i}|{\hat {A}}|\phi _{j}\rangle -\langle {\tilde {\phi }}_{i}|{\hat {A}}|{\tilde {\phi }}_{j}\rangle \right)\langle p_{j}|}$.

ｗhere the indices ${\displaystyle i,j}$ run over all projectors on all atoms. Usually only indices on the same atom are summed over, i.e. off-site contributions are ignored, and this is called the "on-site approximation".

In the original paper, Blöchl notes that there is a degree of freedom in this equation for an arbitrary operator ${\displaystyle {\hat {B}}}$, that is localised inside the spherical augmentation region, to add a term of the form:

${\displaystyle {\hat {B}}-\sum _{i,j}|p_{i}\rangle \langle {\tilde {\phi }}_{i}|{\hat {B}}|{\tilde {\phi }}_{j}\rangle \langle p_{j}|}$,

which can be seen as the basis for implementation of pseudopotentials within PAW, as the nuclear coulomb potential can now be substituted with a smoother one.

• Rostgaard, Carsten (2010). "The Projector Augmented-wave Method". arXiv: [cond-mat.mtrl-sci].
• Kresse, G.; Joubert, D. (1999). "From ultrasoft pseudopotentials to the projector augmented-wave method". Physical Review B. 59 (3): 1758–1775. Bibcode:1999PhRvB..59.1758K. doi:10.1103/PhysRevB.59.1758.
• Dal Corso, Andrea (2010-08-11). "Projector augmented-wave method: Application to relativistic spin-density functional theory". Physical Review B. 82 (7): 075116. Bibcode:2010PhRvB..82g5116D. doi:10.1103/PhysRevB.82.075116.

## Related Research Articles

In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and electrons in a molecule can be treated separately, based on the fact that the nuclei are much heavier than the electrons. The approach is named after Max Born and J. Robert Oppenheimer who proposed it in 1927, in the early period of quantum mechanics.

Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

Coupled cluster (CC) is a numerical technique used for describing many-body systems. Its most common use is as one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry, but it is also used in nuclear physics. Coupled cluster essentially takes the basic Hartree–Fock molecular orbital method and constructs multi-electron wavefunctions using the exponential cluster operator to account for electron correlation. Some of the most accurate calculations for small to medium-sized molecules use this method.

Pseudo-spectral methods, also known as discrete variable representation (DVR) methods, are a class of numerical methods used in applied mathematics and scientific computing for the solution of partial differential equations. They are closely related to spectral methods, but complement the basis by an additional pseudo-spectral basis, which allows representation of functions on a quadrature grid. This simplifies the evaluation of certain operators, and can considerably speed up the calculation when using fast algorithms such as the fast Fourier transform.

Møller–Plesset perturbation theory (MP) is one of several quantum chemistry post–Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by means of Rayleigh–Schrödinger perturbation theory (RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934 by Christian Møller and Milton S. Plesset.

In physics, a pseudopotential or effective potential is used as an approximation for the simplified description of complex systems. Applications include atomic physics and neutron scattering. The pseudopotential approximation was first introduced by Hans Hellmann in 1934.

In mathematical physics, some approaches to quantum field theory are more popular than others. For historical reasons, the Schrödinger representation is less favored than Fock space methods. In the early days of quantum field theory, maintaining symmetries such as Lorentz invariance, displaying them manifestly, and proving renormalisation were of paramount importance. The Schrödinger representation is not manifestly Lorentz invariant and its renormalisability was only shown as recently as the 1980s by Kurt Symanzik (1981).

In quantum chemistry, n-electron valence state perturbation theory (NEVPT) is a perturbative treatment applicable to multireference CASCI-type wavefunctions. It can be considered as a generalization of the well-known second-order Møller–Plesset perturbation theory to multireference Complete Active Space cases. The theory is directly integrated into many quantum chemistry packages such as MOLCAS, Molpro, DALTON, PySCF and ORCA.

In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions, was self-consistent with the initial field, and he thus called his method the self-consistent field method.

Within computational chemistry, the Slater–Condon rules express integrals of one- and two-body operators over wavefunctions constructed as Slater determinants of orthonormal orbitals in terms of the individual orbitals. In doing so, the original integrals involving N-electron wavefunctions are reduced to sums over integrals involving at most two molecular orbitals, or in other words, the original 3N dimensional integral is expressed in terms of many three- and six-dimensional integrals.

A decoherence-free subspace (DFS) is a subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they are a small section of the system Hilbert space where the system is decoupled from the environment and thus its evolution is completely unitary. DFSs can also be characterized as a special class of quantum error correcting codes. In this representation they are passive error-preventing codes since these subspaces are encoded with information that (possibly) won't require any active stabilization methods. These subspaces prevent destructive environmental interactions by isolating quantum information. As such, they are an important subject in quantum computing, where (coherent) control of quantum systems is the desired goal. Decoherence creates problems in this regard by causing loss of coherence between the quantum states of a system and therefore the decay of their interference terms, thus leading to loss of information from the (open) quantum system to the surrounding environment. Since quantum computers cannot be isolated from their environment and information can be lost, the study of DFSs is important for the implementation of quantum computers into the real world.

In quantum chemistry, Brillouin's theorem, proposed by the French physicist Léon Brillouin in 1934, relates to Hartree–Fock wavefunctions. Hartree–Fock, or the self-consistent field method, is a non-relativistic method of generating approximate wavefunctions for a many-bodied quantum system, based on the assumption that each electron is exposed to an average of the positions of all other electrons, and that the solution is a linear combination of pre-specified basis functions.

In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle.

The Peierls substitution method, named after the original work by Rudolf Peierls is a widely employed approximation for describing tightly-bound electrons in the presence of a slowly varying magnetic vector potential.

## References

1. Blöchl, P.E. (1994). "Projector augmented-wave method". Physical Review B . 50 (24): 17953–17978. arXiv:. Bibcode:1994PhRvB..5017953B. doi:10.1103/PhysRevB.50.17953. PMID   9976227.
2. "PAW atomic data for ABINIT code". Archived from the original on 11 September 2015. Retrieved 13 February 2012.
3. "Periodic Table of the Elements for PAW Functions" . Retrieved 13 February 2012.
4. "Atomic PAW Setups" . Retrieved 14 February 2012.
5. Pickard, Chris J.; Mauri, Francesco (2001). "All-electron magnetic response with pseudopotentials: NMR chemical shifts". Physical Review B . 63 (24): 245101–245114. arXiv:. Bibcode:2001PhRvB..63x5101P. doi:10.1103/PhysRevB.63.245101.