2.14. Integral Handling

As the number of nonzero integrals grows very rapidly and reaches easily hundreds of millions even with medium sized basis sets in medium sized molecules, storage of all integrals is not generally feasible. This desperate situation prevented SCF calculations on larger molecules for quite some time, so that Almlöf [169, 170, 171] made the insightful suggestion to repeat the integral calculation, which was already the dominant step, in every SCF cycle to solve the storage problem. Naively, one would think that this raises the effort for the calculation to $n_{\text{iter}}{t}_{\text{integrals}}$ (where $n_{\text{iter}}$ is the number of iterations and $t_{\text{integrals}}$ is the time needed to generate the nonzero integrals). However, this is not the case because only the change in the Fock matrix is required from one iteration to the next, but not the Fock matrix itself. As the calculations starts to converge, more and more integrals can be skipped. The integral calculation time will still dominate the calculation quite strongly, so that ways to reduce this burden are clearly called for. As integrals are calculated in batches[1] the cost of evaluating the given batch of shells $p,q,r,s$ may be estimated as:

(2.82)$$\text{cost}\approx n_p{n}_q{n}_r{n}_s\left(2l_p+1\right)\left(2l_q+1\right)\left(2l_r+1\right)\left(2l_s+1\right)$$

Here, $n_p$ is the number of primitives involved in shell $p$, and $l_p$ is the angular momentum for this shell. Large integrals are also good candidates for storage, because small changes in the density that multiply large integrals are likely to give a nonzero contribution to the changes in the Fock matrix.

ORCA thus features two possibilities for integral handling, which are controlled by the variable SCFMode. In the mode Conventional, all integrals above a given threshold are stored on disk (in a compressed format that saves much disk space). In the mode Direct, all two-electron integrals are recomputed in each iteration.

Two further variables are of importance: In the Conventional mode the program may write enormous amounts of data to disk. To ensure this stays within bounds, the program first performs a so-called “statistics run” that gives a pessimistic estimate of how large the integral files will be. Oftentimes, the program will overestimate the amount of disk space required by a factor of two or more. The maximum amount of disk space that is allowed for the integral files is given by MaxDisk (in Megabytes).

On the other hand, if the integral files in Conventional run are small enough to fit into the central memory, it is faster to do this since it avoids I/O bottlenecks. The maximum amount of memory allocated for integrals in this way is specified by MaxIntMem (in Megabytes). If the integral files are larger than MaxIntMem, no integrals will be read into memory.

%scf
  MaxIter   100    # Max. no. of SCF iterations
  SCFMode Direct   # default, other choice: Conventional
  Thresh  1e-8     # Threshold for neglecting integrals / Fock matrix contributions
                   # Depends on the chosen convergence tolerance (in Eh).
  TCut    1e-10    # Threshold for neglecting primitive batches. If the prefactor
                   # in the integral is smaller than TCut, the contribution of the
                   # primitive batch to the total batch is neglected.
  DirectResetFreq 20   # default: 15
  MaxDisk  2500        # Max. amount of disk for 2 el. ints. (MB)
  MaxIntMem 400        # Max. amount of RAM for 2 el. ints. (MB)
end

The value of DirectResetFreq sets the number of incremental Fock matrix builds after which the program should perform a full Fock matrix build in a Direct SCF calculation. To prevent numerical instabilities that arise from accumulated errors in the recursively build Fock matrix, the value should not be too large, since this will adversely affect the SCF convergence. If the value is too small, the program will update more frequently, but the calculation will take considerably longer, since a full Fock matrix build is more expensive than a recursive one.

The thresholds TCut and Thresh also deserve a closer explanation. Thresh is a threshold that determines when to neglect two-electron integrals. If a given integral is smaller than Thresh Eh, it will not be stored or used in Fock matrix construction. Additionally, contributions to the Fock matrix that are smaller than Thresh Eh will not be calculated in a Direct SCF.

Clearly, it would be wasteful to calculate an integral, then find out it is good for nothing and thus discard it. A useful feature would be an efficient way to estimate the size of the integral before it is even calculated, or even have an estimate that is a rigorous upper bound on the value of the integral. Häser and Ahlrichs [172] were the first to recognize that such an upper bound is actually rather easy to calculate. They showed that:

(2.83)$$\left|⟨ij|kl⟩\right|⩽\sqrt{⟨ij|ij⟩}\sqrt{⟨kl|kl⟩}$$

where:

(2.84)$$⟨ij|kl⟩=\int \int {\varphi }_i\left({\overrightarrow{r}}_1\right){\varphi }_j\left({\overrightarrow{r}}_1\right)r_{12}^{-1}{\varphi }_k\left({\overrightarrow{r}}_2\right){\varphi }_l\left({\overrightarrow{r}}_2\right)d{\overrightarrow{r}}_{1}d{\overrightarrow{r}}_2$$

Thus, in order to compute an upper bound for the integral only the right hand side of this equation must be known. This involves only two index quantities, namely the matrix of two center exchange integrals $⟨ij|ij⟩$. These integrals are easy and quick to calculate and they are all $⩾$0 so that there is no trouble with the square root. Thus, one has a powerful device to avoid computation of small integrals. In an actual calculation, the Schwartz prescreening is not used on the level of individual basis functions but on the level of shell batches because integrals are always calculated in batches. To realize this, the largest exchange integral of a given exchange integral block is looked for and its square root is stored in the so called pre-screening matrix K (that is stored on disk in ORCA). In a Direct SCF this matrix is not recalculated in every cycle, but simply read from disk whenever it is needed. The matrix of exchange integrals on the level of individual basis function is used in Conventional calculations to estimate the disk requirements (the “statistics” run).

Once it has been determined that a given integral batch survives it may be calculated as:

(2.85)$$⟨ij|kl⟩=\sum _p{d}_{pi}\sum _q{d}_{qj}\sum _r{d}_{kr}\sum _s{d}_{sl}⟨i_p{j}_q|k_r{l}_s⟩$$

where the sums $p,q,r,s$ run over the primitive Gaussians in each basis function $i,j,k,l$ and the $d$’s are the contraction coefficients. There are more powerful algorithms than this one and they are also used in ORCA. However, if many terms in the sum can be skipped and the total angular momentum is low, it is still worthwhile to compute contracted integrals in this straightforward way. In equation (2.85), each primitive integral batch $⟨i_p{j}_q|k_r{l}_s⟩$ contains a prefactor $I_{Gaussians}$ that depends on the position of the four Gaussians and their orbital exponents. Since a contracted Gaussian usually has orbital exponents over a rather wide range, it is clear that many of these primitive integral batches will contribute negligibly to the final integral values. In order to reduce the overhead, the parameter TCut is introduced. If the common prefactor $I_{pqrs}$ is smaller than TCut, the primitive integral batch is skipped. However, $I_{pqrs}$ is not a rigorous upper bound to the true value of the primitive integral. Thus, one has to be more conservative with TCut than with Thresh. In practice it appears that choosing TCut=0.01*Thresh provides sufficient accuracy, but the user is encouraged to determine the influence of TCut if it is suspected that the accuracy reached in the integrals is not sufficient.

Hint

• If the direct SCF calculation is close to convergence but fails to finally converge, this maybe related to a numerical problem with the Fock matrix update procedure – the accumulated numerical noise from the update procedure prevents sharp convergence. In this case, set Thresh and TCut lower and/or let the calculation more frequently reset the Fock matrix (DirectResetFreq).

Note

• For a Direct calculation, there is no way to have Thresh larger than TolE. If the errors in the Fock matrix are larger than the requested convergence of the energy, the change in energy can never reach TolE. The program checks for that.

• The actual disk space used for all temporary files may easily be larger than MaxDisk. MaxDisk only pertains to the two-electron integral files. Other disk requirements are not currently checked by the program and appear to be uncritical.

2.14.1. Compression and Storage

The data compression and storage options deserve some comment: in a number of modules including RI-MP2, MDCI, CIS, (D) correction to CIS, etc. the program uses so called “Matrix Containers”. This means that the data to be processed is stored in terms of matrices in files and is accessed by a double label. A typical example is the exchange operator K${}^{\mathbf{\text{ij}}}$ with matrix elements $K^{ij}(a,b)=(ia|jb)$. Here the indices $i$ and $j$ refer to occupied orbitals of the reference state and $a$ and $b$ are empty orbitals of the reference state. Data of this kind may become quite large (formally $N^4$ scaling). To store the numbers in single precision cuts down the memory requirements by a factor of two with (usually very) slight loss in precision. For larger systems one may also gain advantages by also compressing the data (e.g. use a “packed” storage format on disk). This option leads to additional packing/unpacking work and adds some overhead. The simple keywords to control this behavior are given in Table 2.58. For small molecules UCDOUBLE is probably the best option, while for larger molecules UCFLOAT or particularly CFLOAT may be the best choice. Compression does not necessarily slow the calculation down for larger systems since the total I/O load may be substantially reduced and thus (since CPU is much faster than disk) the work of packing and unpacking takes less time than to read much larger files (the packing may reduce disk requirements for larger systems by approximately a factor of 4 but it has not been extensively tested so far). There are many factors contributing to the overall wall clock time in such cases including the total system load. It may thus require some experimentation to find out with which set of options the program runs fastest with.

Caution

• It is possible that FLOAT may lead to unacceptable errors. Thus it is not the recommended option when MP2 or RI-MP2 gradients or relaxed densities are computed. For this reason the default is DOUBLE.

• If you have convinced yourself that FLOAT is OK, it may save you a factor of two in both storage and CPU.

2.14.2. Distance Dependent Pre-Screening

The “Direct” SCF procedure consists of re-calculating the electron-electron repulsion integrals in every SCF iteration. Avoiding to store these integrals is what made SCF calculations on large molecules feasible and therefore is a cornerstone of modern quantum chemistry. The recalculation is only feasible in reasonable turnaround times of the calculation of negligibly small integrals is avoided before calculation. Thus, there needs to be a reasonably cheap to compute estimate for these integrals that programs rely on for avoiding the actual integral calculation. This is the important subject of pre-screening. Pre-screening reduces the number of integrals to be calculated from O(N${}^4$) (exact integrals) or O(N${}^3$) (RI integrals) to O(N${}^2$) because in a large molecule, there are only O(N) significant charge distributions. Of course, it is beneficial if the integral estimate provides a rigorous upper bound to the actual integral because in this case it is guaranteed that no contribution to the Fock or Kohn-Sham matrix that exceeds a pre-defined accuracy threshold is being missed.

The by far most popular integral estimate is the Schwartz estimator introduced by Häser and Ahlrichs:

$${|(\mu \nu |\kappa \tau )|}_{Schwartz}\le \sqrt{\left(\mu \nu |\mu \nu \right)}\sqrt{\left(\kappa \tau |\kappa \tau \right)}$$

Thus, all that is required in order to compute this estimate is the pre-screening matrix

$$Q_{\mu \nu }=\sqrt{\left(\mu \nu |\mu \nu \right)}$$

Which is cheap and not memory consuming. We note in passing that, since integrals are calculated in batches involving four shells, the pre-screening matrix is also shell contracted, meaning the maximum element over the members of both shells is taken as actual pre-screening element. The extension to RI integrals is trivial and will therefore not be written our here.

The Schwartz estimate is used in practically every quantum chemistry program. Despite it’s success, it also has shortcomings. These shortcomings are best understood by considering that a given charge distribution $\mu \nu$ has a multipole structure. At sufficient distance, the interaction of the distributions $\mu \nu$ and $\kappa \tau$ therefore have a distance dependence that depends on the multipolar structure of the two distributions. For example, the monopole-monopole interaction goes as R$-1$ (R being the distance between the centers of the two distributions), dipole-dipole interaction fall off as R${}^{-3}$ etc. Evidently, there is no distance dependence in the Schwartz estimate and consequently, it will have a tendency to strongly overestimate integrals that involve non-negligible but distant charge distributions.

Fig. 2.2 Demonstration how the Schwartz-estimate can drastically overestimate integrals. The red trace Is the Schwartz-estimate while the black trace is the actual interaction of two one-center fs charge distributions as a function of distance.

Historically, a different, related estimator has been popular in the early days of direct SCF. This is the overlap estimator (OVLR-estimate). Consider the two-electron integral:

$$\left(\mu \nu \right|\kappa \tau )={\int }_{}^{}{\int }_{}^{}\frac{\mu ({\mathbf{r}}_1)\nu ({\mathbf{r}}_1)\kappa ({\mathbf{r}}_2)\tau ({\mathbf{r}}_2)}{|{\mathbf{r}}_1-{\mathbf{r}}_2|}d{\mathbf{r}}_{1}d{\mathbf{r}}_2$$

And pretend that we can replace the inverse electronic distance by an effective distance between the two charge distributions:

$$|{\mathbf{r}}_1-{\mathbf{r}}_2|\to R_{eff}$$

In this case, the integral simply becomes:

$$\left(\mu \nu \right|\kappa \tau )=\frac{S_{\mu \nu }{S}_{\kappa \tau }}{R_{eff}}$$

With S being the overlap integral. The estimator is then obtained by taking the maximum absolute element of the overlap matrix elements over the members of the involved shells.

$${|(\mu \nu |\kappa \tau )|}_{OVLR}=\frac{{\tilde{S}}_{\mu \nu }{\tilde{S}}_{\kappa \tau }}{R_{eff}}$$

The effective distance is:

$$R_{eff}=R_{\mu \nu ,\kappa \tau }-{ext}_{\mu \nu }-{ext}_{\mu \nu ,\kappa \tau }$$

The estimator is applied if

$$R_{eff}>1$$

And defaults back to the Schwartz estimate otherwise. In this equation $R_{\mu \nu ,\kappa \tau }=|R_{\mu \nu }-R_{\kappa \tau }|$ is the distance between the centers of the two charge distributions ($R_{\mu \nu }=⟨\mu \vee r\vee \nu ⟩$) and ${ext}_{\mu \nu }$ is the extent of a charge distribution which is discussed in detail in the section on multipole approximations. In a nutshell ${ext}_{\mu \nu }$ defines a radius outside of which the product $\mu (r)\nu (r)$ is zero. Thus, the condition $R_{eff}>1$ is the multipole permissible criterion. If it is met, the charge distributions are non-overlapping and the multipole estimate is convergent.

Based on this realization, Ochsenfeld and co-workers have developed a series of integral estimators that are designed to take this distance dependence into account. In their initial work, Lambrecht and Ochsenfeld provided a concise analysis of the multipolar structure of the charge distribution. After some experimentation, they settled on proposing the QQR-estimator which is a good compromise in terms of computational cost and efficiency in eliminating small contributions:

$${|(\mu \nu |\kappa \tau )|}_{QQR}=\frac{Q_{\mu \nu }{Q}_{\kappa \tau }}{R_{eff}}$$

Which is applied if $R_{eff}>1$, as above. This estimate is not completely rigorous, but still very safe because it assumes monopole moments on the two charge distributions. The Schwartz-integral instead of the overlap integral is assumed to mimic the multipole structure of the charge distributions.

Since experience indicates that the QQR estimator is not eliminating significantly more integrals than the Schwartz-estimate, Thomson and Ochenseld proposed a refined estimator that they refer to as “CSAM”. It is given by:

$${|(\mu \nu |\kappa \tau )|}_{CSAM}=Q_{\mu \nu }{Q}_{\kappa \tau }max(T_{\mu \kappa }{T}_{\nu \tau },T_{\mu \tau }{T}_{\nu \kappa })$$

Where the T-matrix elements are shell maxima of the two-center Coulomb pre-screening integrals:

$$T_{\mu \kappa }=\frac{\sqrt{(\mu \mu |\kappa \kappa )}}{\sqrt{Q_{\mu \mu }{Q}_{\kappa \kappa }}}$$

The T-factors bring in the distance dependence of the estimated integral.

Finally, there is a multipole estimator that takes the lowest multipole moment of each charge distribution into account.

$${|(\mu \nu |\kappa \tau )|}_{multipole}=max\left(\frac{M_{Lmin,m}^{\mu \nu }{M}_{Lmi{n}^{\prime },m^{\prime }}^{\kappa \tau }}{R^{Lmin+Lmi{n}^{{}^{\prime }}+1}}\right)$$

Some idea about the tightness of the estimators can be seen from the following plot:

Fig. 2.3 Tightness of various integral estimators for the model system (glycine)${}_3$ with the def2-SVP basis set.

One can see that the Schwartz-estimator is rigorous and the QQR estimator is nearly rigorous but not very much tighter. The OVLR estimator appears to be the by far tightest estimate but is scattering evenly above and below the correct value. CSAM leads to some underestimation of the integral values and appears to be slightly tighter than QQR. The lowest multipole estimate is ridiculous and of unusable quality.

Whether the revised estimators leads to actual computational savings can be seen in a model calculation on (glycine)${}_{15}$/def2-SVP :

Fig. 2.4 actual computation times and total Hartree-Fock energies for (glycine)${}_{15}$ under the influence of different integral estimators.

The results show that only CSAM leads to savings on the order of 10%, although at the loss of some accuracy which in this example amounts to about 0.01 micro-Eh. The origin of the limited savings are certainly related to the fact that the lowest multipole of a two-center charge distribution is equal to the overlap integral between the two distributions. Thus, as soon as two basis functions have an overlap, they will have a monopole moment and the decay of the interaction between two such charge distributions is only R^-1^ which falls off so slowly that it is not of practical relevance in the elimination of small integrals.

The following input triggers the different estimates:

%shark Prescreening Schwartz
 OVLR
 QQR
 CSAM
 Multipole
End

Note

• In practice OVLR and Multipole are highly unstable and are not recommended

• The integral estimates are NOT yet available for RI integrals.

Relevant Papers:

1. Häser, Marco; Ahlrichs, Reinhart. Improvements on the direct SCF method. Journal of Computational Chemistry, 1989, 10 (1), 104–111. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/jcc.540100111, DOI: 10.1002/jcc.540100111.

2. Lambrecht, Daniel S.; Ochsenfeld, Christian. Multipole-based integral estimates for the rigorous description of distance dependence in two-electron integrals. The Journal of Chemical Physics, 2005, 123 (18), 184101. arXiv:https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.2079967/15373755/184101\_1\_online.pdf, DOI: 10.1063/1.2079967.

3. Thompson, Travis H.; Ochsenfeld, Christian. Distance-including rigorous upper bounds and tight estimates for two-electron integrals over long- and short-range operators. The Journal of Chemical Physics, 2017, 147 (14), 144101. arXiv:https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.4994190/14837196/144101\_1\_online.pdf, DOI: 10.1063/1.4994190.

2.14.3. The “BubblePole” Approximation

The algorithms for Fock matrix construction in ORCA have been optimized to the point where the COSX approximation scales linearly with system size and so does the XC integration. Thus, asymptotically the quadratic scaling of the Coulomb term is dominating the Fock matrix construction time in very large systems. It is well-known that the introduction of hierarchical multipole approximations to the integrals can bring down the scaling of the Coulomb term to linear or near linear. This is the subject of the fast multipole method (FMM) approximation. The FMM approximation has been introduced in ORCA for the calculation of the point charge contribution to embedding calculations and significantly brings down the execution time.

While it may seem tempting to also introduce the FMM approximation in the quantum-quantum charge interactions, the ORCA developers have taken a different route. In FMM algorithms, real space is divided into hierarchical boxes followed by the calculation of the multipole interaction between boxes provided that a multipole allowedness criterion is met.

While it has been proven to work by several authors, it appeared to us that the boxing algorithm is not very natural to chemistry and leads to a number of problems that we intended to avoid in our alternative development.

The Bubblepole (BUPO) approximation is based on a different partitioning that is based on spheres (“Bubbles”). These bubbles fully enclose collections of “quantum objects”. These quantum objects may be shell-pairs, auxiliary basis function shells or also point charges. The criterion for grouping a number of such objects together is spatial proximity. Since shell pairs and auxiliary shells are charge distributions, they have a spatial extent which must be taken into account in the group assembly algorithm. Taking the example of shell pairs, each surviving shell is assigned a shell pair center $R_{\mu \nu }$ and an extent ${ext}_{\mu \nu }$. The precise definition of these quantities is unique to ORCA and are fully discussed in the literature.

The BUPO algorithm then uses a variant of the Kmeans algorithm to group a predefined number of objects (e.g. 150) into each “bottom level” bubble. The bubble center is the arithmetic mean of all enclosed objects and the bubble radius is adjusted such that all objects are fully enclosed together with their extents. This ensures that there is no “leakage” of probability density outside of each bubble.

After setting up the bottom level bubbles, a bubble hierarchy is created in which multipoles are translated from the one bubble layer to the next. Super-bubbles contain lower-level bubbles (typically around three), until at the top layer, there only is a single bubble that encloses the entire molecule. This only happens one initially during system setup).

In the present ORCA implementation, the BUPO algorithm has been combined with ORCA’s most efficient Coulomb-construction algorithm – the Split-RI-J algorithm. In the resulting RI-BUPO-J algorithm, the near-field is treated with the Split-RI-J method and the far-field by the hierarchical construction. The RI-J method generally contains three significant steps

“Projection” of the density on the auxiliary basis set:

$$g_K=\sum _{\mu \nu }^{}{P}_{\mu \nu }\left(\mu \nu |K\right)$$

Solution of linear equations to get the aux-basis density:

$$P_K=\sum _L^{}{\left(V^{-1}\right)}_{KL}{g}_L$$

Assembly of the Coulomb matrix:

$$J_{\mu \nu }=\sum _K^{}{P}_K\left(\mu \nu |K\right)$$

The second step is computationally negligible and proceeds very efficienctly via the Cholesky decomposition. It is not linear scaling but about three to four orders of magnitude cheaper than the other steps such that for all treatable system sizes, it is insignificant. In the first step, the multipole approximation is applied to the density in the orbital basis and therefore the bubbles contain shell-pairs from the basis sets and multipoles derived from the molecular density expanded in the basis set. In the third step, the bubbles contain auxiliary basis shell pairs and multipoles derived from the aux-basis density. The subtleties that derive from this construction are discussed in detail in the original publication.

The numerical results indicate that RI-BUPO-J is asymptotically linear scaling with system size. On linear chains, the crossover with the Split-RI-J algorithm occurs at around 30-40 glycine units. In three dimensional systems, it will be significantly later. Hence, a real advantage of the RI-BUPO/J algorithm over Split-RI-J will only occur for very large systems. This is not due to shortcomings of the BUPO construction but is testimony of the exceptional efficiency of Split-RI-J.

Fig. 2.5 Numerical results for the regular RI-J algorithm, Split-RI-J and RI-BUPO-J on linear glycine chains with basis sets of increasing quality.

There is a significant number of options for BUPO. However, we do NOT recommend to change the defaults. This is expert territory.

Simple Keyword line:

! RI-BUPO/J

Detailed parameter definitions: 

%shark ExtentOpt Ext_SemiNumeric. # default: how to calculate the extents
Ext_analytic
Ext_Numeric
TSphere 1e-15     # shell pair extent cut-off
MP_LAbsMax_BAS 32 # Max L allowed in BAS multipoles
MP_LAbsMin_BAS 0  # Min L allowed in BAS multipoles
MP_Lmax_BAS 10    # Bottom level expansion length
MP_Lincr_BAS 4    # Expansion length increase for
                  # subsequent bubble levels
MP_Rallow_BAS 1.0 # Far field met if effective bubble
                  # distance larger than this (Bohrs)
MP_TScreen_BAS 1e-10  # Multipole elimination threshold
MP_ClusterDim_BAS 150 # Number of objects bottom level bubble
MP_ClusterDim2_BAS 3  # Number of bubbles in super-bubble
                      # in higher level of the hierarchy
MP_NLevels_BAS 15     # Max number of bubble levels in the
                      # hierarchy
# these parameters apply to the first step of RI-BUPO-J
# the same variables exist w.r.t. the auxbasis using the
# postfix \_AUX and apply to the second step of RI-BUPO-J
End

Relevant Papers:

1. Colinet, Pauline; Neese, Frank; Helmich-Paris, Benjamin. Improving the Efficiency of Electrostatic Embedding Using the Fast Multipole Method. J. Comput. Chem., 2025, 46 (1), e27532. DOI: 10.1002/jcc.27532.

2. Neese, Frank; Colinet, Pauline; DeSouza, Bernardo; Helmich-Paris, Benjamin; Wennmohs, Frank; Becker, Ute. The “Bubblepole” (BUPO) Method for Linear-Scaling Coulomb Matrix Construction with or without Density Fitting. J. Phys. Chem. A, 2025, 129 (10), 2618–2637. DOI: 10.1021/acs.jpca.4c07415.

3. White, Christopher A.; Johnson, Benny G.; Gill, Peter M.W.; Head-Gordon, Martin. The continuous fast multipole method. Chemical Physics Letters, 1994, 230 (1), 8–16. DOI: 10.1016/0009-2614(94)01128-1.

2.14.4. Angular Momentum Limits

Energies and gradients in ORCA can now (starting from ORCA 6.1) be performed up to L=10 in the orbital and aux basis sets. Note that the nomenclature changed to the accepted spectroscopic notation:

L=0	1	2	3	4	5	6	7	8	9	10
S	P	D	F	G	H	I	K	L	M	N

Note

• no j-functions! The program will reject it.

• L is reserved for combined s,p shells for historical reasons. If you want to input L=8 functions, simply use the symbol “8” instead of “L”

Here is an example of how to use the angular momentum definition in an example of a user-specified basis set:

! RHF cc-pV6Z VeryTightSCF PrintBasis PModel
%basis

# Basis set for element : Ne
NewGTO Ne
S 11
  1 902400.0000000000 0.0000064569
  2 135100.0000000000 0.0000501790
  3 30750.0000000000 0.0002638325
  4 8710.0000000000 0.0011134543
  5 2842.0000000000 0.0040396234
  6 1026.0000000000 0.0130374676
  7 400.1000000000 0.0377405416
  8 165.9000000000 0.0967943419
  9 72.2100000000 0.2108241400
 10 32.6600000000 0.3586501519
 11 15.2200000000 0.3985795167
S 11
  1 902400.0000000000 -0.0000045125
  2 135100.0000000000 -0.0000351554
  3 30750.0000000000 -0.0001851517
  4 8710.0000000000 -0.0007804845
  5 2842.0000000000 -0.0028452422
  6 1026.0000000000 -0.0092079116
  7 400.1000000000 -0.0271417038
  8 165.9000000000 -0.0715447701
  9 72.2100000000 -0.1678190251
 10 32.6600000000 -0.3267206864
 11 15.2200000000 -0.4994236511
S 1
  1 7.1490000000 1.0000000000
S 1
  1 2.9570000000 1.0000000000
S 1
  1 1.3350000000 1.0000000000
S 1
  1 0.5816000000 1.0000000000
S 1
  1 0.2463000000 1.0000000000
P 5
  1 815.6000000000 0.0014608751
  2 193.3000000000 0.0126013201
  3 62.6000000000 0.0668956161
  4 23.6100000000 0.2559896696
  5 9.7620000000 0.7470043852
P 1
  1 4.2810000000 1.0000000000
P 1
  1 1.9150000000 1.0000000000
P 1
  1 0.8476000000 1.0000000000
P 1
  1 0.3660000000 1.0000000000
P 1
  1 0.1510000000 1.0000000000
D 1
  1 13.3170000000 1.0000000000
D 1
  1 5.8030000000 1.0000000000
D 1
  1 2.5290000000 1.0000000000
D 1
  1 1.1020000000 1.0000000000
D 1
  1 0.4800000000 1.0000000000
F 1
  1 10.3560000000 1.0000000000
F 1
  1 4.5380000000 1.0000000000
F 1
  1 1.9890000000 1.0000000000
F 1
  1 0.8710000000 1.0000000000
G 1
  1 8.3450000000 1.0000000000
G 1
  1 3.4170000000 1.0000000000
G 1
  1 1.3990000000 1.0000000000
H 1
  1 6.5190000000 1.0000000000
H 1
  1 2.4470000000 1.0000000000
I 1
  1 4.4890000000 1.0000000000
K 1
  1 3.0000000000 1.0000000000
# This would be L, but we cannot use the letter L
# because this is used for combined S-P shells
  8 1
  1 3.5000000000 1.0000000000
  M 1
1 2.0000000000 1.0000000000
  N 1
1 1.0000000000 1.0000000000
 end;
end

%shark FockFlag Force_SHARK
       end

* xyz 0 1
Ne 0 0 0
*

2.14.5. Keywords

Table 2.58 Simple input keywords related to integral handling

Keyword	Description
Direct	Selects an integral direct calculation
Conv	Selects an integral conventional calculation
CheapInts	Use the cheap integral feature in direct SCF calculations
NoCheapInts	Turn that feature off
KeepInts	Do not delete the integrals from disk after a calculation in conventional mode
ReadInts	Read the existing integrals from a previous calculation in conventional mode
Float	Set storage format for numbers to single precision (SCF, RI-MP2, CIS, CIS(D), MDCI)
Double	Set storage format for numbers to double precision (default)
UCFloat	Use float storage in the matrix containers without data compression
CFloat	Use float storage in the matrix containers with data compression
UCDouble	Use double storage in the matrix containers without data compression
CDouble	Use double storage in the matrix containers with data compression

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A batch is a set of integrals that arises from all components of the shells involved in the integral. For example a $⟨pp|pp⟩$ batch gives rise to $3\times 3\times 3\times 3=81$ integrals due to all possible combinations of $p_x$, $p_y$ and $p_z$ functions in the four shells. Computations based on batches lead to great computational advantages because the 81 integrals involved in the $⟨pp|pp⟩$ batch share many common intermediate quantities.