Final Report Summary - OF-DFT/MCHF (Orbital Free Density Functional Theory: insights from quantum chemistry and atomic physics)
The aim of the fellowship was to train the fellow in the state-of-the art in density functional theory (DFT), emphasizing both quantum chemistry and atomic physics, with the ultimate goal of using the acquired expertise to develop and implement an orbital free DFT. A key ingredient in this theory is an accurate approximation for the non-interacting kinetic energy functional. The fellow would gain key research experience and acquire a position of scientific independence.
The mobility would allow the fellow to extend his scientific network; by moving to the UK he would improve his communication skills by working with native English speakers. The fellow would acquire competences in transferable skills such as scientific methodology, computer programming, and project management. A roadmap was laid out, describing seven key challenges involving both atomic and molecular systems. The first four were scheduled for year one and the final three for year two. The first two challenges were to acquire expertise with the Wu-Yang method, for computing near-exact DFT quantities that are central to the project (non interacting kinetic energies, orbital energies and exchange-correlation potentials) from accurate electron densities, placing an emphasis on unoccupied Kohn-Sham orbitals. The third and forth challenges were to investigate recently proposed approaches for computing electron affinities (from a corrected Koopmans approach) and the non-interacting kinetic energy (from a single orbital expression).
These first four challenges all had the potential to provide key information to aid the development of an improved non-interacting kinetic energy functional. The final three challenges were the implementation and application of existing approximations to the non-interacting kinetic energy functional and the development of new approximations. To achieve the objectives, the fellow would need to undertake a detailed assessment of the published literature and be trained in quantum chemical software coding and application (using the CADPAC and DALTON programs). To aid dissemination, the fellow would be further trained in scientific writing.
The fellow initially studied the relevant literature and was trained to use the CADPAC and DALTON programs for both general quantum chemical calculations and Wu-Yang calculations. The first phase of the project addressed Challenges 1 and 2. The Wu-Yang (WY) method was used (Challenge 1) to provide key information concerning the exact non-interacting kinetic energy functional (used later in Challenge 7). Specifically, experimental and near-exact WY data were used to determine the effective homogeneity of the exact non-interacting kinetic energy functional under density scaling, placing particular emphasis on the choice of electron affinity and the integer discontinuity in the kinetic potential. Given that the underlying approach could be applied to any functional, it was also decided to broaden the study to include the exact exchange-correlation functional, since this is another key unknown quantity in DFT.
The study indicated that although the exact non-interacting kinetic energy functional is not exactly homogeneous in density scaling, the effective homogeneity is relatively system independent; typical homogeneities are slightly below the anticipated Thomas-Fermi value of 5/3. For the exchange-correlation energy, the effect of the integer discontinuity was much more pronounced, but the average value was strikingly close to the Dirac value of 4/3. This work was published in J. Chem. Phys. 136 034101 (2012). A key aspect of these calculations was to obtain an accurate description of unoccupied Kohn-Sham orbitals; detailed investigations were therefore carried out into the effect of diffuse basis sets on the unoccupied orbitals (Challenge 2).
Next, the project addressed Challenge 3. The exchange-correlation effective homogeneity results were used to provide key insight into the recently proposed corrected Koopmans approach for computing electron affinities. It was demonstrated that enforcing the near exact homogeneity led to an improved correlation with experimental values, but significantly overestimated affinities. Optimal effective homogeneities were therefore determined and a simple scheme was proposed for enforcing an average optimal value. Application of the scheme to a series of organic molecules maintained the excellent correlation with experimental values, whilst significantly reducing the absolute errors. The study also considered the effect of the asymptotic exchange-correlation potential. The work was published in J. Phys. Chem. A. 116, 5497 (2012).
Next, the project addressed Challenges 5, 6, and 7, including the ultimate goal of developing a new non-interacting kinetic energy functional. The earlier homogeneity study had provided near-exact values for the effective homogeneity under density scaling of the non-interacting kinetic energy functional. This was new information that had not previously been used in functional development. The values exhibited limited system dependence, leading us to suggest (Challenge 7) that a functional form should be chosen, whose homogeneity under density scaling was exactly equal to the average near-exact value. As is commonly done, the functional form was also constrained to satisfy the exact coordinate scaling condition. In order to satisfy these two constraints, a generalized gradient approximation form was used; the remaining coefficient was determined from an energy criterion. For the G1 set of molecules, the new functional provided a modest improvement in non-interacting kinetic energies compared to the local functional that did not satisfy the new density scaling condition. More importantly, we also considered whether the new functionals were able to bind simple diatomic molecules.
This is a well-known challenge for orbital free DFT. For CO, F2 and P2, the new functional did successful predict binding, whereas the local functional yielded purely repulsive curves. This is an exciting observation, which suggests that imposing an approximate density scaling condition may be beneficial, offering a new route to improved functionals. In order to put the results from the new functional into context, a wide range of previously published non-interacting kinetic energy functionals were also implemented and tested (Challenges 5, 6). Only a selected few exhibited binding of the diatomic and this could be traced to the fact that these functionals had been determined with an emphasis on accurate forces. The work was submitted to J. Chem. Phys. on 11/11/12.
The fellow has benefited from the hosts scientific network, allowing him to travel to the Norwegian Centre for Theoretical and Computational Chemistry (CTCC) in Oslo in 2011 and 2012 (financed by the CTCC), during which he gave invited seminars on the effective homogeneity (Challenge 1) and electron affinity (Challenge 3) projects. In July 2012, the fellow attended the Sostrup Summer School in Denmark, providing key training in ab initio electronic structure methods, broadening his scientific knowledge. The fellow's skills at dissemination were developed through poster presentations at international conferences in Oslo (Jun 2011) and Ghent (April 2012). In addition to research training, the fellow has also participated in undergraduate teaching in the Chemistry Department at Durham, integrating into the department and developing teaching skills.
In summary, the published homogeneity work provides key new information for the development of both non-interacting kinetic energy functionals and exchange-correlation functionals. The latter has already been used in the current project and the former is currently being investigated. The published electron affinity work will be of significant interest to those working on resonance states. The submitted work on density scaling in non-interacting kinetic energy functionals offers the possibility of a novel, new route to improved functionals, which is essential if the full potential of orbital free DFT is to be realized.
The mobility would allow the fellow to extend his scientific network; by moving to the UK he would improve his communication skills by working with native English speakers. The fellow would acquire competences in transferable skills such as scientific methodology, computer programming, and project management. A roadmap was laid out, describing seven key challenges involving both atomic and molecular systems. The first four were scheduled for year one and the final three for year two. The first two challenges were to acquire expertise with the Wu-Yang method, for computing near-exact DFT quantities that are central to the project (non interacting kinetic energies, orbital energies and exchange-correlation potentials) from accurate electron densities, placing an emphasis on unoccupied Kohn-Sham orbitals. The third and forth challenges were to investigate recently proposed approaches for computing electron affinities (from a corrected Koopmans approach) and the non-interacting kinetic energy (from a single orbital expression).
These first four challenges all had the potential to provide key information to aid the development of an improved non-interacting kinetic energy functional. The final three challenges were the implementation and application of existing approximations to the non-interacting kinetic energy functional and the development of new approximations. To achieve the objectives, the fellow would need to undertake a detailed assessment of the published literature and be trained in quantum chemical software coding and application (using the CADPAC and DALTON programs). To aid dissemination, the fellow would be further trained in scientific writing.
The fellow initially studied the relevant literature and was trained to use the CADPAC and DALTON programs for both general quantum chemical calculations and Wu-Yang calculations. The first phase of the project addressed Challenges 1 and 2. The Wu-Yang (WY) method was used (Challenge 1) to provide key information concerning the exact non-interacting kinetic energy functional (used later in Challenge 7). Specifically, experimental and near-exact WY data were used to determine the effective homogeneity of the exact non-interacting kinetic energy functional under density scaling, placing particular emphasis on the choice of electron affinity and the integer discontinuity in the kinetic potential. Given that the underlying approach could be applied to any functional, it was also decided to broaden the study to include the exact exchange-correlation functional, since this is another key unknown quantity in DFT.
The study indicated that although the exact non-interacting kinetic energy functional is not exactly homogeneous in density scaling, the effective homogeneity is relatively system independent; typical homogeneities are slightly below the anticipated Thomas-Fermi value of 5/3. For the exchange-correlation energy, the effect of the integer discontinuity was much more pronounced, but the average value was strikingly close to the Dirac value of 4/3. This work was published in J. Chem. Phys. 136 034101 (2012). A key aspect of these calculations was to obtain an accurate description of unoccupied Kohn-Sham orbitals; detailed investigations were therefore carried out into the effect of diffuse basis sets on the unoccupied orbitals (Challenge 2).
Next, the project addressed Challenge 3. The exchange-correlation effective homogeneity results were used to provide key insight into the recently proposed corrected Koopmans approach for computing electron affinities. It was demonstrated that enforcing the near exact homogeneity led to an improved correlation with experimental values, but significantly overestimated affinities. Optimal effective homogeneities were therefore determined and a simple scheme was proposed for enforcing an average optimal value. Application of the scheme to a series of organic molecules maintained the excellent correlation with experimental values, whilst significantly reducing the absolute errors. The study also considered the effect of the asymptotic exchange-correlation potential. The work was published in J. Phys. Chem. A. 116, 5497 (2012).
Next, the project addressed Challenges 5, 6, and 7, including the ultimate goal of developing a new non-interacting kinetic energy functional. The earlier homogeneity study had provided near-exact values for the effective homogeneity under density scaling of the non-interacting kinetic energy functional. This was new information that had not previously been used in functional development. The values exhibited limited system dependence, leading us to suggest (Challenge 7) that a functional form should be chosen, whose homogeneity under density scaling was exactly equal to the average near-exact value. As is commonly done, the functional form was also constrained to satisfy the exact coordinate scaling condition. In order to satisfy these two constraints, a generalized gradient approximation form was used; the remaining coefficient was determined from an energy criterion. For the G1 set of molecules, the new functional provided a modest improvement in non-interacting kinetic energies compared to the local functional that did not satisfy the new density scaling condition. More importantly, we also considered whether the new functionals were able to bind simple diatomic molecules.
This is a well-known challenge for orbital free DFT. For CO, F2 and P2, the new functional did successful predict binding, whereas the local functional yielded purely repulsive curves. This is an exciting observation, which suggests that imposing an approximate density scaling condition may be beneficial, offering a new route to improved functionals. In order to put the results from the new functional into context, a wide range of previously published non-interacting kinetic energy functionals were also implemented and tested (Challenges 5, 6). Only a selected few exhibited binding of the diatomic and this could be traced to the fact that these functionals had been determined with an emphasis on accurate forces. The work was submitted to J. Chem. Phys. on 11/11/12.
The fellow has benefited from the hosts scientific network, allowing him to travel to the Norwegian Centre for Theoretical and Computational Chemistry (CTCC) in Oslo in 2011 and 2012 (financed by the CTCC), during which he gave invited seminars on the effective homogeneity (Challenge 1) and electron affinity (Challenge 3) projects. In July 2012, the fellow attended the Sostrup Summer School in Denmark, providing key training in ab initio electronic structure methods, broadening his scientific knowledge. The fellow's skills at dissemination were developed through poster presentations at international conferences in Oslo (Jun 2011) and Ghent (April 2012). In addition to research training, the fellow has also participated in undergraduate teaching in the Chemistry Department at Durham, integrating into the department and developing teaching skills.
In summary, the published homogeneity work provides key new information for the development of both non-interacting kinetic energy functionals and exchange-correlation functionals. The latter has already been used in the current project and the former is currently being investigated. The published electron affinity work will be of significant interest to those working on resonance states. The submitted work on density scaling in non-interacting kinetic energy functionals offers the possibility of a novel, new route to improved functionals, which is essential if the full potential of orbital free DFT is to be realized.