Ccsd Computational Chemistry

[Melchior Dow]

Themethod was initially developed by Fritz Coester and Hermann Kmmel in the 1950s for studying nuclear-physics phenomena, but became more frequently used when in 1966 Jiř Čžek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules. It is now one of the most prevalent methods in quantum chemistry that includes electronic correlation.

CC theory is simply the perturbative variant of the many-electron theory (MET) of Oktay Sinanoğlu, which is the exact (and variational) solution of the many-electron problem, so it was also called "coupled-pair MET (CPMET)". J. Čžek used the correlation function of MET and used Goldstone-type perturbation theory to get the energy expression, while original MET was completely variational. Čžek first developed the linear CPMET and then generalized it to full CPMET in the same work in 1966. He then also performed an application of it on the benzene molecule with Sinanoğlu in the same year. Because MET is somewhat difficult to perform computationally, CC is simpler and thus, in today's computational chemistry, CC is the best variant of MET and gives highly accurate results in comparison to experiments.[4][5][6]

A criticism of the method is that the conventional implementation employing the similarity-transformed Hamiltonian (see below) is not variational, though there are bi-variational and quasi-variational approaches that have been developed since the first implementations of the theory. While the above ansatz for the wave function itself has no natural truncation, however, for other properties, such as energy, there is a natural truncation when examining expectation values, which has its basis in the linked- and connected-cluster theorems, and thus does not suffer from issues such as lack of size extensivity, like the variational configuration-interaction approach.

where T 1 \displaystyle T_1 is the operator of all single excitations, T 2 \displaystyle T_2 is the operator of all double excitations, and so forth. In the formalism of second quantization these excitation operators are expressed as

The exponential operator e T \displaystyle e^T may be expanded as a Taylor series, and if we consider only the T 1 \displaystyle T_1 and T 2 \displaystyle T_2 cluster operators of T \displaystyle T , we can write

The resulting similarity-transformed Hamiltonian is non-Hermitian, resulting in different left and right vectors (wave functions) for the same state of interest (this is what is often referred to in coupled-cluster theory as the biorthogonality of the solution, or wave function, though it also applies to other non-Hermitian theories as well). The resulting equations are a set of non-linear equations, which are solved in an iterative manner. Standard quantum-chemistry packages (GAMESS (US), NWChem, ACES II, etc.) solve the coupled-cluster equations using the Jacobi method and direct inversion of the iterative subspace (DIIS) extrapolation of the t-amplitudes to accelerate convergence.

The classification of traditional coupled-cluster methods rests on the highest number of excitations allowed in the definition of T \displaystyle T . The abbreviations for coupled-cluster methods usually begin with the letters "CC" (for "coupled cluster") followed by

One possible improvement to the standard coupled-cluster approach is to add terms linear in the interelectronic distances through methods such as CCSD-R12. This improves the treatment of dynamical electron correlation by satisfying the Kato cusp condition and accelerates convergence with respect to the orbital basis set. Unfortunately, R12 methods invoke the resolution of the identity, which requires a relatively large basis set in order to be a good approximation.

Considering the fact that the CC method was well understood around the late fifties[,] it looks strange that nothing happened with it until 1966, as Jiř Čžek published his first paper on a quantum chemistry problem. He had looked into the 1957 and 1960 papers published in Nuclear Physics by Fritz and myself. I always found it quite remarkable that a quantum chemist would open an issue of a nuclear physics journal. I myself at the time had almost given up the CC method as not tractable and, of course, I never looked into the quantum chemistry journals. The result was that I learnt about Jiř's work as late as in the early seventies, when he sent me a big parcel with reprints of the many papers he and Joe Paldus had written until then.

Josef Paldus also wrote his first-hand account of the origins of coupled-cluster theory, its implementation, and exploitation in electronic wave-function determination; his account is primarily about the making of coupled-cluster theory rather than about the theory itself.[15]

are related to the cluster operators T \displaystyle T , since in the limit of including up to T N \displaystyle T_N in the cluster operator the CC theory must be equal to full CI, we obtain the following relationships[16][17]

In nuclear physics, coupled cluster saw significantly less use than in quantum chemistry during the 1980s and 1990s. More powerful computers, as well as advances in theory (such as the inclusion of three-nucleon interactions), have spawned renewed interest in the method since then, and it has been successfully applied to neutron-rich and medium-mass nuclei. Coupled cluster is one of several ab initio methods in nuclear physics and is specifically suitable for nuclei having closed or nearly closed shells.[21]

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For the introductory course for the students I am trying to collect the brief overview of different quantum chemistry methods and their numerical complexity. The second point is surprisingly poorly explained in all textbooks I was able to find. In the best case, it is a small note like

[source is "Parallel computing in quantum chemistry" by C.L. Janssen, 2008], but there are no estimates for all methods discussed in the book and I have problems to find this kind of information elsewhere.

Because different computational methods require somewhat different expertise, it seems to me that, for each and every method, complexity/computational cost is commented separately. Therefore the exact details needs to be investigated maybe from books dedicated to a particular computational method. But of course there are some experts in the field who collectively reported these data.

... the scaling of other popular Quantum Chemistry methods, which ranges from $O(N^4)$ for Hartree Fock (HF) to $O(N^5)$ for MP2, $O(N^6)$ for MP3, and $O(N^7)$ for MP4, CISD(T) and CCSD(T), it still makes large scale simulations prohibitive.

It's probably more useful to know why these methods come with the aforementioned cost scalings. It's all about identifying the rate-limiting step in the calculation, which is typically a type of tensor contraction. Let's take Hartree-Fock for example. The most costly step comes from the construction of the effective one-body interaction, which for a conventional closed-shell calculation, looks like this

We can identify the first term in $g_\mu\nu$ as the Coulomb interaction and the second term as exchange. If one were to unravel the above tensor contraction to execute using some nested do/for loops, one would need something like this (using Fortran 90 as an example syntax):

Clearly, the above loop has a computational expense that scales as $\mathcalO(N^4)$ because we have 4 nested loops, each running from $1$ to $N$. Another example we can do is CCSD. If you derive the CCSD equations (in spinorbital form for simplicity), one finds that the most expensive term is the so-called ladder diagram contribution $\frac14\sum_mnef v_mn^ef t_ef^ij t_ab^mn$ where $v_pq^rs$ are the antisymmetrized electron repulsion integrals and $t_ab^ij$ are the 2-body cluster amplitudes. If we were to repeat the above procedure of naively unravelling into nested "do" loops, one find that the ladder diagram has a cost that scales as $\mathcalO(N^8)$. Why then is CCSD listed as $\mathcalO(N^6)$? The answer is that the tensor contraction can be broken up into two steps. If one first computes $\chi_mn^ij = \sum_ef v_mn^ef t_ef^ij$, this operation scales as $\mathcalO(N^6)$. Then, one computes $\sum_mn \chi_mn^ij t_ab^mn$ which also scales as $\mathcalO(N^6)$. Thus, you can compute the $\mathcalO(N^8)$ contraction at the cost of no more than multiple $\mathcalO(N^6)$ operations, hence CCSD scales as $\mathcalO(N^6)$! These types of time-saving factorizations are easily deduced by inspection of the diagrammatic form of these many-body equations (or, one can just look at the tensor equations directly, however, it's a bit harder that way). Optimal factorization is of utmost importance in any quantum chemistry code.

And you can also reduce the scaling of MP2, TDHF, and HF using density-fitting and Cholesky decompositions by an order of magnitude to $\cal O(N^4)$ and $\cal O(N^3)$, respectively. There are also density-fitting type speedups (called least-squares tensor hypercontraction, or LS-THC) that can be applied to coupled-cluster that reduce the scaling of CCSD to $\cal O(N^4)$ which is probably the best you can get. Certainly, every method can have its scaling reduced dramatically by introducing some kind of orbital localization because the steep scaling of quantum chemistry stems from the fact that even atoms far away from each other, say on opposite ends of a molecule, end up interacting because the standard MO basis used is delocalized. However, localization comes with a heavy cost of losing orthonormality, so it's generally tricky implement. Actually, a very promising class of methods are the rank-compression methods like Cholesky decomposition because you can very robustly reduce the time needed for rate-limiting tensor contractions while having a very controllable and systematic error - something most other "fast scaling" approaches do not have.

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