Predicting Molecular and Electronic Response to Magnetic Field from First Principles
Predicting Molecular and Electronic Response to Magnetic Field from First Principles
Alfredo Correa, Eunseok Lee, Giulia Galli and Wei Cai
Abstract
[sections will be erased in PRL format, they are here to organize the editing]
Introduction
Electromagnetic field interaction with matter is a primary way to probe matter.
Ab initio molecular dynamics under magnetic field (non-perturbatively) is still impossible.
If enabled, AIMD under magnetic field will allow us to predict a wide range of effects, including NMR, structural change of molecules, optical effects (band gap change).
Progress in applying electric field (non-perturbatively) to insultator.
In the case of electric field the only non-perturbative approach requieres the use of the Berry phase [Resta, etc].
Progress in applying magnetic field through MPBC [Cai04, Lee].
The progress since the PRL 04 paper. (1) Generalize to more elements. (2) Powerful to study large/dynamical systems.
It was limited to local pseudopotential, which effectively limited us to Hydrogen. It is now generalized to non-local pseudopotential, magnetic translation operator. (The generalized Bloch theorem may be included in this paper.) Consistent formulation of current operator in the presence of non-local pseudopotential.
Formulation of current operator in the presence of magnetic field and non-local operators.
The formulation was limited to local pseudopotentials which effectively limited to one element Hydrogen. We have now removed this constraint and are able to study a wide range of elements. This requires us to reformulate the translation operator in the presence of magnetic field and non-local operators.
It was implemented in a toy code. Not powerful enough to handle large/dynamic systems. Now implemented on large scale (mainstream) Qbox code. Now we can do AIMD simulations. For the first time, directly compute/observe the effect of magnetic field on molecular and electronic structure on matter.
In this Letter, we take advantage of this progress to study something. Specific findings of this paper. Compute NMR for CH4. Trend in the equation of state, optical gap, orientation of molecules w.r.t. magnetic field.
Now that we put all of these together, this presents an unified description (toolbox) of molecules and condensed matter under magnetic field.
Method
The development that lead to this.
Single-electron Hamiltonian with magnetic field with density functional theory. V includes hartree and DFT part. Kinetic energy includes vector potential. By solving this equation self-consistently, we get energy of the system and electron density.
Under zero magnetic field, translation operators commute with with Hamiltonian and with themselves -- leads to Bloch's theorem. This is the basis of plane-wave codes, which are the most widely used method. Advantages of PBC.
Under magnetic field, original translation operators do not commute with Hamiltonian. New translation operator has to be defined, which commute with Hamiltonian. But they do not (in general) commute with themselves, unless magnetic flux is quantized. In this case, we restored Bloch theorem, but with a different set of translation operators.
The (new) Bloch's theorem results in MPBC --- in the language of magnetic translation operators. We retain the advantages of plane-wave codes. FFT.
(we finished the review of MPBC by now)
Why do we need pseudopotentials. Plane-wave methods require us to remove core electrons, which requires pseudopotential. Non-local psedopotential is needed for accurate predictions for most elements H. We limit our discussions to pseudopotentials of the Kleinman-Bylander form. V = |phi><phi|.
What is different if we have a non-local pseudopotential in the Hamiltonian? For the Bloch's theorem to hold (magnetic translation operator commutes with Hamiltonian), we need to make |phi> an eignefunction of the magnetic translation operator.
This means we should not simply multiply a structural factor in the Reciprocal space. Instead, we multiply part of the structural factor in the Intermediate space, and part in the Reciprocal space.
(why do we care about the current?)
Under zero magnetic field, the electron current density is usually zero due to time-reversal symmetry.
Under magnetic field, time-reversal symmetry is broken, which induces electronic current. This leads to an induced magnetic field. The coupling of the induced magnetic field with the spin of the nuclei is responsible for the NMR signal.
The current density operator J in the presence of magnetic field and non-local pseudopotential includes the commutator Vnl with r. What's non-trivial? Current operator now depends on the problem. This enables us to directly compute NMR signals for all nuclei in the simulation cell. (more direct method).
Results
We implemented all of these nice algorithms (formulations) in a large-scale ab initio program Qbox [Ref].
NMR
Previous approaches to compute NMR.
Perturbative approach. Cumbersome in both theory and implementation.
Converse approach -- apply localized magnetic field to each nucleus. A separate calculation is needed to find NMR signal for each nucleus.
MPBC provides a direct method to obtain NMR signals by applying a finite magnetic field to the entire simulation cell. Electron wave functions now contains a current. How do we compute it. Sum over all occupied states, apply the current operator.
Plot of induced current flow around CH4 molecule. Table of compute NMR signal with experiments and previous methods.
(Right now we still have some bug in the calculation of current. The current around H and CH4 does not look symmetrical. For Hydrogen, we can use KSSOLV to double check Qbox results.)
(Perhaps the normalization factor for Jx (relative to Jy and Jz). Try different cell sizes, energy cut-off values. Jx, Jy, Jz should converge. )
Orientation of Molecules in Magnetic Field
Magnetic field not only changes the details of electronic structure. It can also change the structure of the molecule. For example, in the same CH4 molecule above, CH4 energy depends on its orientation of the magnetic field.
Energy of CH4 molecule as a function of orientation. Polar plot?
AIMD of Some Fluid
(could we do without it?)
Born-Oppenheimer approximation. Electron at ground state. Ions move classically.
Admit that trajectory is not real, because we lack Berry phase forces [Ref]. But thermodynamics is correct. Theorem: the energy of a classical system of ions cannot depend on magnetic field [van Leeuwen, see Martin p.578].
D2 or H2O (some person claimed magnetic field affects property of water. Alfredo will look for the reference)
g(r) with/without magnetic field. Averaged band-gap. Does the pressure change with magnetic field.
(Is the Qbox code ready to do MD? Is the force correct?)