Dec 202014
 
  • Tensor product methods and entanglement optimization for ab initio quantum chemistry. This review of tensor product approximations in quantum chemistry by Szalay et al. was just posted on arXiv. At times a bit formal, this is a nice introduction to the topic.
  • In Search of a Rational Dressing of Intermediate Effective Hamiltonians. This is a paper on intermediate Hamiltonian theory, an approach to deal with the intruder state problem in effective Hamiltonian theories. From the abstract: The intermediate effective Hamiltonians are designed to provide M exact energies and the components of the corresponding eigenvectors in the N-dimensional model space, with N > M. The effective Hamiltonian is not entirely defined by these N × M conditions, and several dressings of the Hamiltonian matrix in the model space are possible.
  • Appointing silver and bronze standards for noncovalent interactions: A comparison of spin-component-scaled (SCS), explicitly correlated (F12), and specialized wavefunction approaches. A systematic investigation of the accuracy of non-covalent interaction predicted with various quantum chemical methods by Burns and co-workers. From the abstract: After examination of both accuracy and performance for 394 model chemistries, SCS(MI)-MP2/cc-pVQZ can be recommended for general use, having good accuracy at low cost and no ill-effects such as imbalance between hydrogen-bonding and dispersion-dominated systems or non-parallelity across dissociation curves. Moreover, when benchmarking accuracy is desirable but gold-standard computations are unaffordable, this work recommends silver-standard [DW-CCSD(T**)-F12/aug-cc-pVDZ] and bronze-standard [MP2C-F12/aug-cc-pVDZ] model chemistries, which support accuracies of 0.05 and 0.16 kcal/mol and efficiencies of 97.3 and 5.5 h for adenine·thymine, respectively.
 December 20, 2014  Uncategorized No Responses »
Oct 262014
 

9781493912766_p0_v2_s260x420
I recently purchased a couple dozen books for the group and out of curiosity decided to add this book to the order. Although I don’t use integrals much in my research (we mostly deal with linear algebra and tensor algebra) I found this book to be extremely interesting. It’s so interesting that I cannot stop reading it and put it down.
Inside Interesting Integrals is a great book if you want to revisit your integration skills and learn all sorts of tricks to compute definite integrals.

 October 26, 2014  Uncategorized No Responses »
Oct 042014
 

In the past two weeks I read two papers from arXiv that present some very interesting new ideas.  The first paper, Extended Møller-Plesset perturbation theory for dynamical and static correlations, by Takashi Tsuchimochi and Troy Van Voorhis, deals with symmetry restoration in second-order perturbation theory.  The extended MP2 method presented by the authors looks promising and I found it very interesting because it can automatically adapt to single- and multireference problems.  At the very end of the paper the authors even show that their technique can be used to compute excited states.

The second one,Compact wavefunctions from compressed imaginary time evolution, is from Jarrod R. McClean and Alán Aspuru-Guzik.  This paper mixes compression techniques with a propagation of the Schrödinger equation in imaginary time to compute the ground state energy.

 

 October 4, 2014  Uncategorized No Responses »
Mar 042014
 

Our new computer cluster, Helium, arrived yesterday!

This little stack of metal and silicon packs 20 nodes with 2 Intel Xeon E5-2650 v2 CPUs and 128 GB of RAM, connected by a QDR Infiniband switch (for a total of 320 cores). Helium is a cluster in progress. Once RAM is cheaper we are going to add a few more nodes (12) with 256 GB of memory.

Helium packed

As you can see it barely fit into our elevators.

Helium size

photo 4

Wallace and Francesco building the cluster.

photo 2-1

Front and rear. Check out our QDR Infiniband switch!

photo 3-1
photo 4-1

 March 4, 2014  Uncategorized 3 Responses »
Feb 172014
 

I just created a web site for the 2014 meeting of the Southeast Theoretical Chemistry Association (SETCA). This year this event will be hosted at Emory and I will be the principal organizer. Information about the meeting can be found at www.setca2014.info.

 February 17, 2014  Uncategorized No Responses »
Sep 162013
 

Just found this interesting paper on the structure of the simplest Criegee intermediate. The Criegee has been for a long time an elusive molecule and pops up in many fields like astrochemistry and atmospheric chemistry.

Communication: Determination of the molecular structure of the simplest Criegee intermediate CH2OO
J. Chem. Phys. 139, 101103 (2013)
http://dx.doi.org/10.1063/1.4821165
 September 16, 2013  Uncategorized No Responses »
Aug 212013
 

Just read this very interesting article on compression of the wave function information using singular value decomposition of the full CI vector. The idea of factorizing the CI matrix can already be found in a paper by Koch and Dalgaard, however this earlier paper showed results for system with small number of electrons and basis sets and has a different purpose. As mentioned in the paper, the cost of the SVD step used in the compression step scales as N_{\rm FCI}^{3/2}, where N_{\rm FCI} is the size of the FCI space. This step actually costs a bit more (asymptotically) than what is required to evaluate the product of the Hamiltonian matrix times a trial vector.

Lossless compression of wave function information using matrix factorization: A “gzip” for quantum chemistry
J. Chem. Phys. 139, 074113 (2013)
http://dx.doi.org/10.1063/1.4816769
 August 21, 2013  Uncategorized No Responses »
Jun 232013
 

A general quadratic programming problem consists in minimizing a quadratic function of the form: 

    \[f(\mathbf{x}) = \frac{1}{2} \mathbf{x}^{T} \mathbf{Q} \mathbf{x} + \mathbf{c}^{T} \mathbf{x},\]

where the vector \mathbf{x} \in \mathbb{R}^n is subject to inequality and equality constraints:

    \[\mathbf{A} \mathbf{x} \leq \mathbf{b},\]

    \[\mathbf{E} \mathbf{x} = \mathbf{d}.\]

The n \times n matrix \mathbf{Q} is symmetric, while \mathbf{A} and \mathbf{E} are general matrices that enforce the constraints.

It turns out that there are a few interesting problems in electronic structure theory that can be formulated as quadratic programming problems.  For example, the energy of a Slater determinant is: 

    \[E = \sum_{i}^{\rm occ} h_{ii} + \frac{1}{2} \sum_{ij}^{\rm occ} (J_{ij} - K_{ij}),\]

where h_{ii} is the diagonal matrix element of the one-body component of the Hamiltonian, while J_{ij} and K_{ij} are Coulomb and exchange integrals, respectively.  All the sums run over the occupied orbitals of the Slater determinant.

If we introduce the occupation vector \mathbf{n}, with elements n_i = 0,1, then the energy of a Slater determinant may be written as a quadratic function: 

    \[E(\mathbf{n}) = \mathbf{h}^T \mathbf{n} + \frac{1}{2} \mathbf{n}^{T} \mathbf{V} \mathbf{n},\]

 with the matrix \mathbf{V} is defined as V_{ij} = J_{ij}-K_{ij}.  Now let’s ask the question: given a set of molecular orbitals \phi_p, what is the occupation vector that minimizes the energy of a Slater determinant with a given number of alpha and beta electrons (N_\alpha, N_\beta?  This is a binary quadratic programming problem, because the set of conditions given by the general quadratic programming problem are augmented with the requirement that n_i = 0 \text{ or } 1.  Binary quadratic problems are NP-hard, but in the case of the electronic structure counterpart, a good dose of heuristics (physics) can be used to find a solution.

For a wave function whose two- and higher-body cumulants are zero, in the natural orbital basis the energy expression is identical to the  one for a single Slater determinant, with the difference that the natural occupation numbers n_i must satisfy 0 \leq n_i \leq 1.  Thus, finding the optimal uncorrelated state is equivalent to a standard quadratic programming problem.  Interestingly, the computational complexity of this problem appears to depend on the properties of the matrix \mathbf{V} (\mathbf{Q}).  If \mathbf{V} is positive definite this problem can be solved in polynomial time, otherwise it is NP-hard (according to the wikipedia article on quadratic programming).  I do not know whether the \mathbf{V} that arises in electronic structure problems is always positive definite, but I suspect it is.  It would be interesting to check this property numerically.  In addition, if the orbitals are variationally optimal, then n_i will either be 0 or 1, and the uncorrelated state will be a Slater determinant.

 June 23, 2013  Uncategorized No Responses »