THE JOURNAL OF CHEMICAL PHYSICS 142, 064108 (2015)
Non-orthogonal spin-adaptation of coupled cluster methods: A new implementation of methods including quadruple excitations Devin A. Matthews1,a) and John F. Stanton2 1
Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA 2 Department of Chemistry and Biochemistry, Institute for Theoretical Chemistry, University of Texas at Austin, Austin, Texas 78712, USA
(Received 30 October 2014; accepted 20 January 2015; published online 11 February 2015) The theory of non-orthogonal spin-adaptation for closed-shell molecular systems is applied to coupled cluster methods with quadruple excitations (CCSDTQ). Calculations at this level of detail are of critical importance in describing the properties of molecular systems to an accuracy which can meet or exceed modern experimental techniques. Such calculations are of significant (and growing) importance in such fields as thermodynamics, kinetics, and atomic and molecular spectroscopies. With respect to the implementation of CCSDTQ and related methods, we show that there are significant advantages to non-orthogonal spin-adaption with respect to simplification and factorization of the working equations and to creating an efficient implementation. The resulting algorithm is implemented in the CFOUR program suite for CCSDT, CCSDTQ, and various approximate methods (CCSD(T), CC3, CCSDT-n, and CCSDT(Q)). C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907278]
I. INTRODUCTION
Among the most successful electronic structure methods today are those based on the coupled cluster approximation.1–6 This approximation is especially useful for several reasons such as its rigorous size-extensivity, its formulation as a hierarchy of methods with increasing accuracy (and cost)— coupled cluster singles and doubles (CCSD), triples (CCSDT), quadruples (CCSDTQ), and so on—and the fact that the highest rung of this hierarchy for a given molecule is equivalent to full configuration interaction (CI) (i.e., it is exact in the nonrelativistic Born-Oppenheimer framework). Out of this general family, CCSD(T)7 has been so successful as to have been called the “gold standard” of electronic structure theory (as tokened by Dunning). However, the success of CCSD(T) is not universal. For calculations which require extreme accuracy— especially in the range of what is commonly called “subchemical” accuracy (e.g., 100 orbitals) can be computed in a matter of minutes. The most expensive calculation in this table (HSOH with a triple-ζ basis, no molecular symmetry) requires less than 2 h for 86 correlated orbitals. In fact, the limit on the size of the calculations performed in this table is due to the fact that these results were compared directly to existing CCSDT(Q) programs, and these calculations ran as long as 2 days for the (Q) correction. Overall, the CCSDT and CCSDTQ timings for NCC are 5-10 times faster than previously possible, and the (Q) correction can be obtained 20-100 times faster. The difference between the CCSDTQ and CCSDT(Q) improvements highlights the effect of tensor transposition and other data movement on the computation efficiency, as these operations are relatively more important for CCSDT(Q) (which has only an O(n9) floating-point cost compared to O(n8) data cost vs. O(n10) to O(n8) for CCSDTQ). For triples methods, the current code is only somewhat faster than the previous code of Gauss and Stanton, who used non-orthogonal spin adaptation techniques, but chose not to publish the details. In addition, they made no effort to minimize the transposition count, as has been done here. More expensive CCSDT(Q) calculations have also been carried out, at a scale for which it is not feasible to compare to existing codes. Several example results are given in Table III. The benzene dimer calculations were performed using all 12 cores of a dual Xeon X5670 system with five disks in a RAID0 configuration. The other calculations were performed using 4 cores on a Xeon E5-1620 system with only a single hard drive. The latter calculations, owing to the less-performant I/O system, spent more than 50% of the calculation time performing I/O operations. However, even with the modest computational resources allocated, a CCSDT(Q) calculation on a system with more than 150 orbitals can be completed in only a few days. The iterative coupled cluster calculations typically require on the order of 20-30 iterations to reach a convergence level of 10−9 (i.e., the maximum change in the T1 and T2 amplitudes is below this threshold). Thus, the CCSDTQ calculations
presented here represent from about 3 h to about 5 days for the total time-to-solution. This is, in fact, on the short side of typical “large” calculations (which may run for as much as several months), which shows that, in fact, even larger CCSDTQ calculations should be possible using the new code. For the CCSDT(Q) calculations, there are both an iterative and a non-iterative part, but using the timings presented and the typical number of iterations, it is easy to see that the noniterative (Q) correction dominates the computation time. Thus, the timings presented already approximately reflect the total time-to-solution, on the order of minutes to three days for the longest calculation. This is also not the “upper bound” for possible calculations, especially considering possible distributed parallelization of the code and techniques such as frozen virtual natural orbitals (FVNOs).62 All of the methods implemented in NCC are multithreaded explicitly using the OpenMP interface and implicitly through the ability to use a multithreaded linear algebra library for elementary matrix operations. The choice of explicit or implicit threading for a given tensor contraction is determined dynamically by the available parallelism at each level. An example of the parallel speedup obtained through multithreading on a dual Xeon E5620 system is given in Figure 7. The speedup obtained using all 8 cores is about 4×, giving a parallel efficiency of ∼50%. While this is not perfect, it is encouraging given the fact that the code makes no attempt to address issues such as NUMA memory accesses, thread locality, and cache sharing. Also, scheduling of work units (individual matrix multiplications in the explicit threading case) is determined statically, leaving the possibility for increased performance
FIG. 7. Parallel speedup obtained through explicit (OpenMP) and implicit multithreading.
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through dynamic scheduling. The parallel efficiency on one processor only (up to 4 cores) remains at least 75%. V. CONCLUSIONS AND FUTURE WORK
The theory of non-orthogonal spin-adaptation, especially as it is applied to triple and quadruple excitations in coupled cluster theory, has been presented. The working equations using this approach are shown to be highly compact and efficient. A new computer implementation of CCSDT, CCSDT(Q), and CCSDTQ as a part of the CFOUR program suite has been presented and shown to be highly efficient, as much as 10-100 times faster than prior implementations. Finally, the comprehensive computational algorithm utilizing non-orthogonally spin-adapted amplitudes, integrals, and intermediates, which enables this level of efficiency, has been detailed, along with several important optimizations which address computational concerns outside of the traditional focus on floating point operations. The current implementation is limited to single-point energy calculations. Work is underway to extend this to gradients and to excited state calculations using the same theoretical basis (and indeed re-using much of the same code). Additionally, further improvements to efficiency and new types of optimizations are being developed. In particular, the framework provided by the new BLAS-like Library Instantiation Software (BLIS) library63 is being leveraged to enable efficient computation of tensor contractions which do not conform to a simple decomposition into matrix multiplication operations, which should enable a further reduction in the amount of tensor transpositions required. Distributed parallelism is also being explored for CCSDT(Q) and CCSDTQ. Finally, while the program structure and optimizations detailed here are especially suited to non-orthogonal spinadaptation, they are not exclusive to it. Conventional spinintegrated coupled cluster methods can also be implemented in terms of this structure and are expected to show a similar level of efficiency, without the need for completely rewriting the program. Extension of the current code to open-shell systems using this possibility is planned for the near future.
ACKNOWLEDGMENTS
D.A.M. was supported by the US Department of Energy Computational Science Graduate Fellowship, Grant No. DEFG02-97ER25308. This work was also supported by grants from the Welch Foundation (No. F-1283) and the US National Science Foundation (No. CHE-1361031).
APPENDIX: DIAGRAMMATIC RULES FOR NON-ORTHOGONAL SPIN-ADAPTATION
The diagrammatic rules for non-orthogonally spinadapted coupled cluster, using the spin-summed amplitudes and integrals, are given in Table IV along with the corresponding rules in the usual Brandow representation. The main difference between the two sets of rules is that while the Brandow rules are concerned with single indices, the spin-summed rules primarily deal in “pairs,” i.e., columns of indices when the quantities are written as tensors. Diagrammatically, a pair is two lines which connect to a vertex at the same point. Pairs may be either contracted (when both labels are contracted), partially contracted (only one label is contracted), or external. External pairs of the result may also be considered, which are two external lines connected through a sequence of contracted (internal) lines. Additionally, the rules are specialized to binary contractions (diagrams with two vertices), but this is not a restriction as factorization can always be done at the diagrammatic stage and the contractions can be done pairwise. Finally, diagrams interpreted in terms of orbital quantities, even when topologically equivalent to the same Brandow diagram, are not interchangeable since a factor of 2 is applied for each closed loop (and the number of such loops is influenced by topologically indistinguishable permutations). Thus, while it is not necessary to enumerate all such diagrams as in the Goldstone case, the Brandow diagram which is used as the “template” much be chosen carefully. The diagram which must be used is the one which (1) has external lines connected as external pairs consistent with the desired labeling of the orbital result and (2) has the maximal number of closed loops. This diagram is often unique (when it is not,
TABLE IV. Brandow and spin-summed rules for diagram interpretation. Rules are numbered sequentially 1-6. Brandow rules
Spin-summed rules
l-ou t -l abel s . Each vertex A contributes a spin-orbital quantity A al al l-i n-l abel s
Each set of n identical contracted lines gives a factor of
1 n! .
The overall sign is equal to (−1)l+h , where l is the number of closed loops and h is the number of hole lines. Each set of n identical vertices gives a factor of
1 n! .
l-ou t -l abel s . One or both Each vertex A contributes an orbital quantity Aˇ al al l-i n-l abel s vertices are spin-summed (details in text). 1 Each set of n identical external or contracted pairs gives a factor of n! . External pairs must both go in on one vertex and both go out on another (possibly different) vertex.
No change.
No change. This factor must be applied before factorization into binary diagrams, however.
Contracted lines are summed over all spin-orbitals.
Contracted lines are summed over all spatial orbitals.
External lines of the same type which appear on different vertices are antisymmetrized.
All external pairs on the result of the same type are symmetrized.
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all possible compliant diagrams must be used), and indeed, is often the form one would draw without such restrictions. The spin-summation required by the first spin-summed rule can take one of several forms depending on the topology of the diagram. If neither vertex has an external pair (i.e., both lines are uncontracted), then all fully contracted pairs are spinsummed on one of the vertices (either may be chosen). If only one vertex has an external pair, then all fully contracted pairs are spin-summed on that vertex. If both vertices have external pairs, then the contracted pairs are spin-summed on one vertex, the external pair is spin-summed on the other, and a factor of 21 is added. Additionally, a second term is added with a factor of 1 2 , no spin-summation, and where one of the contracted labels and an uncontracted label are interchanged on each vertex. If both vertices have an external label of the same type, then these should be the ones interchanged, and a third term should be added which is identical to the second except now with a factor of 1 and with the additional interchange of the two external labels which were interchanged with the contracted label. This third term is related to the second by a constant factor and a simple permutation of labels, and so both terms can be computed from the same contraction. The remaining rules are then applied to each of the terms generated (this must be done independently since the effect is not the same for the additional terms resulting from the spin-summation step). The effect of rule 3 on any additional terms from rule 1 is always opposite in sign from first term, which can be seen when these additional terms are written as Goldstone diagrams. The spin-summation rule is perhaps best illustrated by a few example cases. First, consider the coupled cluster energy diagrams given in Figure 8. The “correct” diagrams are those which have two closed loops (since there are no external lines, this is the only important feature). For the T2 diagram, rule 1 falls under the case of no external pairs on either vertex. Thus, we can choose to spin-sum the contracted pairs on either vertex. For consistency with the second diagram, we choose to spin-sum the integrals. Rule 2 gives a factor of 12 since there are two identical contracted pairs. Rule 3 gives a positive sign, while rule 6 has no effect. Finally, rule 5 gives a summation over spatial orbitals for the e, f , m, and n labels. The final result
FIG. 8. Coupled cluster energy diagrams with both the “correct” and “incorrect” topologies for interpretation with the spin-summed rules. The slash on the lower-left diagram gives the partitioning into pairwise diagrams. The left T1 vertex along with the integrals is taken as an intermediate F fn .
J. Chem. Phys. 142, 064108 (2015)
is then ( ) f 1 mˇ nˇ e f 2ˇvefmn − vˇ fmne tˇemn , vˇeˇ fˇ tˇmn = 2 efmn efmn
(A1)
which is of course the usual result for closed-shell CCSD.64–66 The diagram involving T1 amplitudes contains three vertices and needs to be factorized. Since there are two equivalent T1 vertices, the final contribution should have a factor of 12 from rule 4. But, factorizing the diagram breaks the symmetry of the T1 vertices. This highlights that the factors arising from equivalent vertices should be evaluated before and incorporated into the factorization since they are not apparent at the two-vertex diagram level. We can factor this diagram as shown in Figure 8, where the slash denotes the partitioning into two diagrams, by first contracting the integrals with the left T1 vertex to give an intermediate Fˇfn . Looking at just this part of the diagram, the integral vertex now has an external pair ( f n), and so by rule 1, we should spin-sum it over the contracted pair em. The remaining rules indicate a positive sign and a sum over e and m, giving the result for the intermediate, ˇ ˇe Fˇfn = vˇemn (A2) ˇ f t m. em
The energy contribution is then given by a two-vertex diagram where this intermediate is fully contracted with a second T1 vertex. Since the diagram is fully contracted, we can spin-sum either vertex (it makes no difference here since either choice simply gives a factor of 2), and the result of the other rules is simply a sum over f and n. Remembering to add in the factor of 21 for the equivalent T1 vertices, we have ) 1 ˇ nˇ f 1 mˇ nˇ e f ( mn Ffˇ tˇn = vˇeˇ fˇ tˇmtˇn = 2ˇvef − vˇ fmne tˇemtˇnf , 2 fn 2 efmn efmn
(A3)
which is again identical to the usual closed-shell CCSD result. A second example is the contribution of T4 to T2 from CCSDTQ, whose diagram is given in Figure 9. This diagram has many Goldstone variants which must be summed and factorized to give an efficient equation. However, application of the spin-summed rules is straightforward here. Specifically, since the T4 vertex has external pairs and the integrals do not, we spin-sum the em and f n pairs on T4. Rule 2 gives a factor of 14 : a factor of 12 each for the two equivalent external and contracted pairs. Rule 3 gives a positive factor and rule 4 has no effect. Rule 5 gives a sum over e, f , m, and( n. Finally, ) rule 6 now comes into play, giving a symmetrizer 1 + Pbjai where Pbjai simultaneously interchanges a with b and i with j. The final
FIG. 9. Topologically “correct” diagram for the contribution of T4 to T2. The (numerous) “incorrect” diagrams are not shown.
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needed (since exchanging i with m breaks the ai-bj symmetry). The third term is the same except without the factor of 12 and interchanging i and k, ( )( ) 1 bj ai mc abe mc ˇ − 1 + Pck + Pck 1 + Pbjai . tˇabe v ˇ + t v ˇ mjk ie 2 m ji ke em (A6)
FIG. 10. Topologically “correct” diagram for the ring part of T3 → T3.
result is then, ) ab eˇ fˇ 1( tˇi j mˇ nˇ vˇefmn. 1 + Pbjai 4 efmn
(A4)
The interested and diligent reader is encouraged to confirm that expanding the spin-orbital equation in terms of orbital quantities does indeed give the same result. This example illustrates the utility of the spin-summed rules; in that, one can go directly from a single familiar diagram to a compact, efficient working equation. To our knowledge, this technique is the most useful approach yet developed for the identification of efficient formulation of closed-shell amplitude equations. eˇ fˇ Since tˇiab jm ˇ nˇ is already symmetric in ai and bj, the symmetrizer and one of the factors of 21 could be canceled. However, when combining individual terms into complete working equations, it is generally advantageous to keep all of the terms with a consistent symmetrizer. In the CCSD case, the orbital equations are often written with extraneous symmetrizers removed, i.e., as tˇijab = {symmetric terms} ( ) + 1 + Pbjai {non-symmetric terms} instead of as ) (1 ( tˇijab = 1 + Pbjai {symmetric terms} 2 ) + {non-symmetric terms} . For CCSD, the difference between these two representations is minimal. But for CCSDT and especially CCSDTQ, there are many types of partial symmetries and hence many different types of symmetrizers which must be applied. Using a single, consistent symmetrizer and adjusting the factors of the individual terms give a significantly cleaner representation. The final example is the T3 “ring” contribution in CCSDT, whose diagram is given in Figure 10. Both vertices have external pairs now, which invokes the third case of rule 1. Thus, the first term has an additional factor of 12 and spin-summation of the em pair on the first vertex and the ck pair on the second vertex. Combined with the additional factor of 21 from rule 2 and the summation and symmetrizer from rules 5 and 6, this term is )( ) 1( bj ai eˇ m cˇ 1 + Pck + Pck 1 + Pbjai tˇiab (A5) jm ˇ vˇe kˇ . 4 em The second term required by rule 1 needs an external label from each vertex interchanged with a contracted label, for example, we can interchange i and k with m and e, respectively. The spin-summation is removed, the sign from rule 3 is now negative, and the additional factor of 12 from rule 3 is no longer
This gives the total contribution, ( )( ) bj ai 1 + Pck + Pck 1 + Pbjai ( ) 1 1 eˇ m cˇ i ˇabe mc × tˇiab v ˇ − + P t v ˇ . m ji ke k 4 j mˇ e kˇ 2 em 1J.
(A7)
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Non-orthogonal spin-adaptation of coupled cluster methods: A new implementation of methods including quadruple excitations Devin A. Matthews1,a) and John F. Stanton2 1
Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA 2 Department of Chemistry and Biochemistry, Institute for Theoretical Chemistry, University of Texas at Austin, Austin, Texas 78712, USA
(Received 30 October 2014; accepted 20 January 2015; published online 11 February 2015) The theory of non-orthogonal spin-adaptation for closed-shell molecular systems is applied to coupled cluster methods with quadruple excitations (CCSDTQ). Calculations at this level of detail are of critical importance in describing the properties of molecular systems to an accuracy which can meet or exceed modern experimental techniques. Such calculations are of significant (and growing) importance in such fields as thermodynamics, kinetics, and atomic and molecular spectroscopies. With respect to the implementation of CCSDTQ and related methods, we show that there are significant advantages to non-orthogonal spin-adaption with respect to simplification and factorization of the working equations and to creating an efficient implementation. The resulting algorithm is implemented in the CFOUR program suite for CCSDT, CCSDTQ, and various approximate methods (CCSD(T), CC3, CCSDT-n, and CCSDT(Q)). C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907278]
I. INTRODUCTION
Among the most successful electronic structure methods today are those based on the coupled cluster approximation.1–6 This approximation is especially useful for several reasons such as its rigorous size-extensivity, its formulation as a hierarchy of methods with increasing accuracy (and cost)— coupled cluster singles and doubles (CCSD), triples (CCSDT), quadruples (CCSDTQ), and so on—and the fact that the highest rung of this hierarchy for a given molecule is equivalent to full configuration interaction (CI) (i.e., it is exact in the nonrelativistic Born-Oppenheimer framework). Out of this general family, CCSD(T)7 has been so successful as to have been called the “gold standard” of electronic structure theory (as tokened by Dunning). However, the success of CCSD(T) is not universal. For calculations which require extreme accuracy— especially in the range of what is commonly called “subchemical” accuracy (e.g., 100 orbitals) can be computed in a matter of minutes. The most expensive calculation in this table (HSOH with a triple-ζ basis, no molecular symmetry) requires less than 2 h for 86 correlated orbitals. In fact, the limit on the size of the calculations performed in this table is due to the fact that these results were compared directly to existing CCSDT(Q) programs, and these calculations ran as long as 2 days for the (Q) correction. Overall, the CCSDT and CCSDTQ timings for NCC are 5-10 times faster than previously possible, and the (Q) correction can be obtained 20-100 times faster. The difference between the CCSDTQ and CCSDT(Q) improvements highlights the effect of tensor transposition and other data movement on the computation efficiency, as these operations are relatively more important for CCSDT(Q) (which has only an O(n9) floating-point cost compared to O(n8) data cost vs. O(n10) to O(n8) for CCSDTQ). For triples methods, the current code is only somewhat faster than the previous code of Gauss and Stanton, who used non-orthogonal spin adaptation techniques, but chose not to publish the details. In addition, they made no effort to minimize the transposition count, as has been done here. More expensive CCSDT(Q) calculations have also been carried out, at a scale for which it is not feasible to compare to existing codes. Several example results are given in Table III. The benzene dimer calculations were performed using all 12 cores of a dual Xeon X5670 system with five disks in a RAID0 configuration. The other calculations were performed using 4 cores on a Xeon E5-1620 system with only a single hard drive. The latter calculations, owing to the less-performant I/O system, spent more than 50% of the calculation time performing I/O operations. However, even with the modest computational resources allocated, a CCSDT(Q) calculation on a system with more than 150 orbitals can be completed in only a few days. The iterative coupled cluster calculations typically require on the order of 20-30 iterations to reach a convergence level of 10−9 (i.e., the maximum change in the T1 and T2 amplitudes is below this threshold). Thus, the CCSDTQ calculations
presented here represent from about 3 h to about 5 days for the total time-to-solution. This is, in fact, on the short side of typical “large” calculations (which may run for as much as several months), which shows that, in fact, even larger CCSDTQ calculations should be possible using the new code. For the CCSDT(Q) calculations, there are both an iterative and a non-iterative part, but using the timings presented and the typical number of iterations, it is easy to see that the noniterative (Q) correction dominates the computation time. Thus, the timings presented already approximately reflect the total time-to-solution, on the order of minutes to three days for the longest calculation. This is also not the “upper bound” for possible calculations, especially considering possible distributed parallelization of the code and techniques such as frozen virtual natural orbitals (FVNOs).62 All of the methods implemented in NCC are multithreaded explicitly using the OpenMP interface and implicitly through the ability to use a multithreaded linear algebra library for elementary matrix operations. The choice of explicit or implicit threading for a given tensor contraction is determined dynamically by the available parallelism at each level. An example of the parallel speedup obtained through multithreading on a dual Xeon E5620 system is given in Figure 7. The speedup obtained using all 8 cores is about 4×, giving a parallel efficiency of ∼50%. While this is not perfect, it is encouraging given the fact that the code makes no attempt to address issues such as NUMA memory accesses, thread locality, and cache sharing. Also, scheduling of work units (individual matrix multiplications in the explicit threading case) is determined statically, leaving the possibility for increased performance
FIG. 7. Parallel speedup obtained through explicit (OpenMP) and implicit multithreading.
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through dynamic scheduling. The parallel efficiency on one processor only (up to 4 cores) remains at least 75%. V. CONCLUSIONS AND FUTURE WORK
The theory of non-orthogonal spin-adaptation, especially as it is applied to triple and quadruple excitations in coupled cluster theory, has been presented. The working equations using this approach are shown to be highly compact and efficient. A new computer implementation of CCSDT, CCSDT(Q), and CCSDTQ as a part of the CFOUR program suite has been presented and shown to be highly efficient, as much as 10-100 times faster than prior implementations. Finally, the comprehensive computational algorithm utilizing non-orthogonally spin-adapted amplitudes, integrals, and intermediates, which enables this level of efficiency, has been detailed, along with several important optimizations which address computational concerns outside of the traditional focus on floating point operations. The current implementation is limited to single-point energy calculations. Work is underway to extend this to gradients and to excited state calculations using the same theoretical basis (and indeed re-using much of the same code). Additionally, further improvements to efficiency and new types of optimizations are being developed. In particular, the framework provided by the new BLAS-like Library Instantiation Software (BLIS) library63 is being leveraged to enable efficient computation of tensor contractions which do not conform to a simple decomposition into matrix multiplication operations, which should enable a further reduction in the amount of tensor transpositions required. Distributed parallelism is also being explored for CCSDT(Q) and CCSDTQ. Finally, while the program structure and optimizations detailed here are especially suited to non-orthogonal spinadaptation, they are not exclusive to it. Conventional spinintegrated coupled cluster methods can also be implemented in terms of this structure and are expected to show a similar level of efficiency, without the need for completely rewriting the program. Extension of the current code to open-shell systems using this possibility is planned for the near future.
ACKNOWLEDGMENTS
D.A.M. was supported by the US Department of Energy Computational Science Graduate Fellowship, Grant No. DEFG02-97ER25308. This work was also supported by grants from the Welch Foundation (No. F-1283) and the US National Science Foundation (No. CHE-1361031).
APPENDIX: DIAGRAMMATIC RULES FOR NON-ORTHOGONAL SPIN-ADAPTATION
The diagrammatic rules for non-orthogonally spinadapted coupled cluster, using the spin-summed amplitudes and integrals, are given in Table IV along with the corresponding rules in the usual Brandow representation. The main difference between the two sets of rules is that while the Brandow rules are concerned with single indices, the spin-summed rules primarily deal in “pairs,” i.e., columns of indices when the quantities are written as tensors. Diagrammatically, a pair is two lines which connect to a vertex at the same point. Pairs may be either contracted (when both labels are contracted), partially contracted (only one label is contracted), or external. External pairs of the result may also be considered, which are two external lines connected through a sequence of contracted (internal) lines. Additionally, the rules are specialized to binary contractions (diagrams with two vertices), but this is not a restriction as factorization can always be done at the diagrammatic stage and the contractions can be done pairwise. Finally, diagrams interpreted in terms of orbital quantities, even when topologically equivalent to the same Brandow diagram, are not interchangeable since a factor of 2 is applied for each closed loop (and the number of such loops is influenced by topologically indistinguishable permutations). Thus, while it is not necessary to enumerate all such diagrams as in the Goldstone case, the Brandow diagram which is used as the “template” much be chosen carefully. The diagram which must be used is the one which (1) has external lines connected as external pairs consistent with the desired labeling of the orbital result and (2) has the maximal number of closed loops. This diagram is often unique (when it is not,
TABLE IV. Brandow and spin-summed rules for diagram interpretation. Rules are numbered sequentially 1-6. Brandow rules
Spin-summed rules
l-ou t -l abel s . Each vertex A contributes a spin-orbital quantity A al al l-i n-l abel s
Each set of n identical contracted lines gives a factor of
1 n! .
The overall sign is equal to (−1)l+h , where l is the number of closed loops and h is the number of hole lines. Each set of n identical vertices gives a factor of
1 n! .
l-ou t -l abel s . One or both Each vertex A contributes an orbital quantity Aˇ al al l-i n-l abel s vertices are spin-summed (details in text). 1 Each set of n identical external or contracted pairs gives a factor of n! . External pairs must both go in on one vertex and both go out on another (possibly different) vertex.
No change.
No change. This factor must be applied before factorization into binary diagrams, however.
Contracted lines are summed over all spin-orbitals.
Contracted lines are summed over all spatial orbitals.
External lines of the same type which appear on different vertices are antisymmetrized.
All external pairs on the result of the same type are symmetrized.
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all possible compliant diagrams must be used), and indeed, is often the form one would draw without such restrictions. The spin-summation required by the first spin-summed rule can take one of several forms depending on the topology of the diagram. If neither vertex has an external pair (i.e., both lines are uncontracted), then all fully contracted pairs are spinsummed on one of the vertices (either may be chosen). If only one vertex has an external pair, then all fully contracted pairs are spin-summed on that vertex. If both vertices have external pairs, then the contracted pairs are spin-summed on one vertex, the external pair is spin-summed on the other, and a factor of 21 is added. Additionally, a second term is added with a factor of 1 2 , no spin-summation, and where one of the contracted labels and an uncontracted label are interchanged on each vertex. If both vertices have an external label of the same type, then these should be the ones interchanged, and a third term should be added which is identical to the second except now with a factor of 1 and with the additional interchange of the two external labels which were interchanged with the contracted label. This third term is related to the second by a constant factor and a simple permutation of labels, and so both terms can be computed from the same contraction. The remaining rules are then applied to each of the terms generated (this must be done independently since the effect is not the same for the additional terms resulting from the spin-summation step). The effect of rule 3 on any additional terms from rule 1 is always opposite in sign from first term, which can be seen when these additional terms are written as Goldstone diagrams. The spin-summation rule is perhaps best illustrated by a few example cases. First, consider the coupled cluster energy diagrams given in Figure 8. The “correct” diagrams are those which have two closed loops (since there are no external lines, this is the only important feature). For the T2 diagram, rule 1 falls under the case of no external pairs on either vertex. Thus, we can choose to spin-sum the contracted pairs on either vertex. For consistency with the second diagram, we choose to spin-sum the integrals. Rule 2 gives a factor of 12 since there are two identical contracted pairs. Rule 3 gives a positive sign, while rule 6 has no effect. Finally, rule 5 gives a summation over spatial orbitals for the e, f , m, and n labels. The final result
FIG. 8. Coupled cluster energy diagrams with both the “correct” and “incorrect” topologies for interpretation with the spin-summed rules. The slash on the lower-left diagram gives the partitioning into pairwise diagrams. The left T1 vertex along with the integrals is taken as an intermediate F fn .
J. Chem. Phys. 142, 064108 (2015)
is then ( ) f 1 mˇ nˇ e f 2ˇvefmn − vˇ fmne tˇemn , vˇeˇ fˇ tˇmn = 2 efmn efmn
(A1)
which is of course the usual result for closed-shell CCSD.64–66 The diagram involving T1 amplitudes contains three vertices and needs to be factorized. Since there are two equivalent T1 vertices, the final contribution should have a factor of 12 from rule 4. But, factorizing the diagram breaks the symmetry of the T1 vertices. This highlights that the factors arising from equivalent vertices should be evaluated before and incorporated into the factorization since they are not apparent at the two-vertex diagram level. We can factor this diagram as shown in Figure 8, where the slash denotes the partitioning into two diagrams, by first contracting the integrals with the left T1 vertex to give an intermediate Fˇfn . Looking at just this part of the diagram, the integral vertex now has an external pair ( f n), and so by rule 1, we should spin-sum it over the contracted pair em. The remaining rules indicate a positive sign and a sum over e and m, giving the result for the intermediate, ˇ ˇe Fˇfn = vˇemn (A2) ˇ f t m. em
The energy contribution is then given by a two-vertex diagram where this intermediate is fully contracted with a second T1 vertex. Since the diagram is fully contracted, we can spin-sum either vertex (it makes no difference here since either choice simply gives a factor of 2), and the result of the other rules is simply a sum over f and n. Remembering to add in the factor of 21 for the equivalent T1 vertices, we have ) 1 ˇ nˇ f 1 mˇ nˇ e f ( mn Ffˇ tˇn = vˇeˇ fˇ tˇmtˇn = 2ˇvef − vˇ fmne tˇemtˇnf , 2 fn 2 efmn efmn
(A3)
which is again identical to the usual closed-shell CCSD result. A second example is the contribution of T4 to T2 from CCSDTQ, whose diagram is given in Figure 9. This diagram has many Goldstone variants which must be summed and factorized to give an efficient equation. However, application of the spin-summed rules is straightforward here. Specifically, since the T4 vertex has external pairs and the integrals do not, we spin-sum the em and f n pairs on T4. Rule 2 gives a factor of 14 : a factor of 12 each for the two equivalent external and contracted pairs. Rule 3 gives a positive factor and rule 4 has no effect. Rule 5 gives a sum over e, f , m, and( n. Finally, ) rule 6 now comes into play, giving a symmetrizer 1 + Pbjai where Pbjai simultaneously interchanges a with b and i with j. The final
FIG. 9. Topologically “correct” diagram for the contribution of T4 to T2. The (numerous) “incorrect” diagrams are not shown.
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needed (since exchanging i with m breaks the ai-bj symmetry). The third term is the same except without the factor of 12 and interchanging i and k, ( )( ) 1 bj ai mc abe mc ˇ − 1 + Pck + Pck 1 + Pbjai . tˇabe v ˇ + t v ˇ mjk ie 2 m ji ke em (A6)
FIG. 10. Topologically “correct” diagram for the ring part of T3 → T3.
result is then, ) ab eˇ fˇ 1( tˇi j mˇ nˇ vˇefmn. 1 + Pbjai 4 efmn
(A4)
The interested and diligent reader is encouraged to confirm that expanding the spin-orbital equation in terms of orbital quantities does indeed give the same result. This example illustrates the utility of the spin-summed rules; in that, one can go directly from a single familiar diagram to a compact, efficient working equation. To our knowledge, this technique is the most useful approach yet developed for the identification of efficient formulation of closed-shell amplitude equations. eˇ fˇ Since tˇiab jm ˇ nˇ is already symmetric in ai and bj, the symmetrizer and one of the factors of 21 could be canceled. However, when combining individual terms into complete working equations, it is generally advantageous to keep all of the terms with a consistent symmetrizer. In the CCSD case, the orbital equations are often written with extraneous symmetrizers removed, i.e., as tˇijab = {symmetric terms} ( ) + 1 + Pbjai {non-symmetric terms} instead of as ) (1 ( tˇijab = 1 + Pbjai {symmetric terms} 2 ) + {non-symmetric terms} . For CCSD, the difference between these two representations is minimal. But for CCSDT and especially CCSDTQ, there are many types of partial symmetries and hence many different types of symmetrizers which must be applied. Using a single, consistent symmetrizer and adjusting the factors of the individual terms give a significantly cleaner representation. The final example is the T3 “ring” contribution in CCSDT, whose diagram is given in Figure 10. Both vertices have external pairs now, which invokes the third case of rule 1. Thus, the first term has an additional factor of 12 and spin-summation of the em pair on the first vertex and the ck pair on the second vertex. Combined with the additional factor of 21 from rule 2 and the summation and symmetrizer from rules 5 and 6, this term is )( ) 1( bj ai eˇ m cˇ 1 + Pck + Pck 1 + Pbjai tˇiab (A5) jm ˇ vˇe kˇ . 4 em The second term required by rule 1 needs an external label from each vertex interchanged with a contracted label, for example, we can interchange i and k with m and e, respectively. The spin-summation is removed, the sign from rule 3 is now negative, and the additional factor of 12 from rule 3 is no longer
This gives the total contribution, ( )( ) bj ai 1 + Pck + Pck 1 + Pbjai ( ) 1 1 eˇ m cˇ i ˇabe mc × tˇiab v ˇ − + P t v ˇ . m ji ke k 4 j mˇ e kˇ 2 em 1J.
(A7)
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