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Combining Pseudopotential and All Electron Density Functional Theory for the Efficient Calculation of Core Spectra using a Multiresolution Approach Laura E. Ratcliff,∗,y,z W. Scott Thornton,{ Alvaro´ V´azquezMayagoitia,x,z and Nichols A. Romerox,z yDepartment of Materials, Imperial College London, London SW7 2AZ, UK zArgonne Leadership Computing Facility, Argonne National Laboratory, Illinois 60439, USA {Stony Brook University, Stony Brook, New York 11794, USA xComputational Science Division, Argonne National Laboratory, Illinois 60439, USA E-mail: laura.ratcliff[email protected] Abstract Introduction Broadly speaking, the calculation of core spec- madness (Multiresolution ADaptive Numeri- tra such as electron energy loss spectra (EELS) cal Environment for Scientific Simulation)1,2 is at the level of density functional theory (DFT) a general purpose numerical framework which usually relies one of two approaches: concep- combines a multiresolution approach with a tually more complex but computationally ef- parallel programming environment designed for ficient projector augmented wave based ap- petascale performance.3 The use of an adaptive proaches, or more straightforward but com- multiresolution approach allows integral and putationally more intensive all electron (AE) differential equations in many dimensions to be based approaches. In this work we present an solved with guaranteed precision. Furthermore, alternative method, which aims to find a mid- the code has been structured so that develop- dle ground between the two. Specifically, we ers can focus on the high level implementation have implemented an approach in the multi- of new functionalities, without needing detailed wavelet madness molecular DFT code which knowledge of the low level technicalities. These permits a combination of atoms treated at the features have facilitated the development of var- AE and pseudopotential (PSP) level. Atoms for ious scientific applications using madness, in- arXiv:1811.08397v2 [physics.chem-ph] 7 May 2019 which one wishes to calculate the core edges are cluding a molecular density functional theory thus treated at an AE level, while the remainder (DFT) code4{6, and many other applications can be treated at the PSP level. This is made spanning a range of fields, many of which are possible thanks to the multiresolution approach concerned with quantum chemistry7{19. of madness, which permits accurate and effi- The molecular DFT code (hereafter referred cient calculations at both the AE and PSP level. to as moldft) provides a setup for the pre- Through examples of a small molecule and a cise treatment of electronic systems with an carbon nanotube we demonstrate the potential excellent cost to accuracy ratio. Nonetheless, applications of our approach. the cost of treating heavy atoms { or large molecules containing many light atoms { re- 1 mains high, inhibiting the ability to study tech- behind the madness molecular DFT code, be- nologically interesting systems such as those fore briefly discussing the implementation of a containing transition metals. The most popular mixed AE/PSP approach. We then present way to alleviate this problem is the replacement the method used to calculate ELNES, includ- of the exact all electron (AE) atomic potential ing the calculation of the virtual Kohn Sham with a smoother pseudopotential (PSP), which (KS) eigenstates. Finally, we validate the ap- has the dual advantage of reducing the number proach through examples of two systems which of electrons requiring explicit treatment (as the benefit from a mixed PSP/AE representation. core electrons are incorporated implicitly), and reducing the number of basis functions needed to represent the wavefunctions close to the nu- Theory clei, thanks to increased smoothness. This en- ables heavy atoms to be treated with a much Molecular DFT with MADNESS lower computational cost and would extend the Both the madness code as a whole and the applicability of moldft. moldft code in particular have already been Beyond these advantages, the multiresolution described elsewhere2,4{6, and so here we give approach of madness also facilitates the im- only an outline. One of the central compo- plementation of a mixed representation, namely nents of madness is the use of a disjoint \multi- the treatment of only select atoms with PSPs, wavelet" basis set, which is constructed from a with others retaining the full AE potential. set of (shifted and scaled) Legendre polynomi- While this is also possible in Gaussian basis sets als represented in a non-uniform grid. The grid, using effective core potentials (ECP), a mul- therefore the basis, is dynamically adapted to tiresolution approach permits a similar method give higher resolution where needed (e.g. close using a systematic basis, since the automatic to the atoms where the KS wavefunctions are refinement ensures that both full and pseudo- more rapidly varying), giving rise to a computa- atoms are treated at a resolution which is high tionally efficient yet highly accurate multireso- enough to maintain accuracy without loss of ef- lution analysis (MRA). Indeed, each KS orbital ficiency. has its own individual adaptively refined repre- One application where this is of particular sentation. The exploitation of MRA techniques use is core level spectroscopy, e.g. electron en- allows the code to reach a finite arbitrary pre- ergy loss spectra (EELS), specifically energy cision, while relieving the user of the need to loss near edge structure (ELNES), which is the manually converge the basis set. region of the spectrum immediately after the In contrast to other DFT codes, the central edge onset. ELNES is able to probe the local eigenvalue equation to be solved is recast from electronic structure of a sample and is thus an a differential equation of the form invaluable experimental technique. Theoretical calculations (often based on DFT) are required 1 − r2 + V (x) (x) = E (x) ; (1) to help interpret experimental spectra. The de- 2 velopment and application of methods for sim- ulating ELNES is therefore an active area of re- to an integral equation of the form 20 search . However, one is often only interested Z in calculating spectra for a subset of a system, (x) = −2 dx −∇2 − 2E−1 V (x) (x) : e.g. a group of atoms or single atomic species (2) within a molecule, or a molecule in an environ- This form is well-suited to the madness frame- ment. In such cases, core states need only be work and has the advantage of being able to be explicitly calculated for the atoms of interest, solved iteratively without the need for a precon- which can easily be achieved by treating only ditioner. In practice, an initial guess for the KS these atoms at the AE level. wavefunctions is first generated by projecting In this paper we first outline the key concepts 2 an atomic orbital basis, typically 6-31G, into well in the wavelet-based BigDFT code28, for the multiwavelet basis. This approach takes ad- calculations in both open and periodic bound- vantage of the underlying madness numerical ary conditions. This also provides a means of and parallel runtime to efficiently solve the KS validating our implementation. equations to a guaranteed precision, while re- The implementation in madness was rela- quiring only minimal input from the user con- tively straightforward, since the underlying ma- cerning the basis set or parallel setup. chinery used to solve the KS equations remains A wide range of LDA, GGA and hybrid the same, only the definition of the atomic po- functionals are available in moldft through the tential need be modified. In the first instance LibXC library21. Although we have not ex- we have not implemented either relativistic ef- ploited the capability in this work, in addi- fects or non-linear core corrections, which have tion to standard canonical orbitals, moldft has demonstrated an accuracy of similar quality to the option to use localized molecular orbitals18, AE calculations27. In the future this might eas- where the Pipek-Mezy scheme22 is used to lo- ily be extended. calize the orbitals. This gives rise to a quasi- linear scaling, reducing the computational cost Mixed AE/PSP Calculations of treating larger systems. As discussed, an adaptive multi-wavelet ap- proach is also highly suitable for a mixed Pseudopotentials in MADNESS AE/PSP representation. Both the implemen- Despite the advantages of an MRA approach tation and application of such an approach is for DFT calculations, the computational cost of straightforward. Consider two opposing sce- treating heavy atoms or indeed of large systems narios: calculations using PWs and those using containing light elements is high, due in part to Gaussian-type basis sets. For the former, aside the explicit treatment of core electrons. The ap- from the prohibitive cost of AE calculations in plicability of moldft could therefore be extended PW basis sets, the delocalized nature of the ba- to new materials by implementing PSPs. Al- sis functions also prevents them from being spa- though most commonly used in periodic plane tially varied to be more or less dense around wave (PW) DFT codes, PSPs are also effec- different atoms. Any mixed AE/PSP approach tively employed in combination with other ba- would therefore also necessitate a mixed basis sis sets. Given such extensive use for DFT cal- set approach, e.g. combining PWs with a local- 29,30 culations, it is of no surprise that PSPs exist ized basis set . in a number of varieties, generally categorized In the case of Gaussian basis sets, the number as \hard" or \soft", depending on the smooth- of functions associated with each atom could ness of the pseudized-wavefunctions. Indeed, be directly modified depending on the level the development of ever cheaper and more ac- of theory used. Such a mixed representation curate PSPs continues to be an active area of has previously been implemented using Gaus- research. It is also worth mentioning the pro- sian basis sets and used to assess the accu- jector augmented wave (PAW) approach23, an racy of individual PSPs for molecular proper- alternative approach to more traditional PSPs, ties including binding energies and vibrational 31 which aims to reproduce the correct AE be- properties .

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