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iCI: iterative CI toward full CI Wenjian Liu, and Mark R. Hoffmann J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.5b01099 • Publication Date (Web): 14 Jan 2016 Downloaded from http://pubs.acs.org on January 15, 2016

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Journal of Chemical Theory and Computation

iCI: iterative CI toward full CI Wenjian Liu∗,† and Mark R. Hoffmann∗,‡ Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and Molecular Engineering, and Center for Computational Science and Engineering, Peking University, Beijing 100871, People’s Republic of China, and Chemistry Department, University of North Dakota, Grand Forks, ND 58202-9024, U.S.A. E-mail: [email protected]; [email protected]

Abstract It is shown both theoretically and numerically that the minimal multireference configuration interaction (CI) approach [Theor. Chem. Acc. 133, 1481 (2014)] converges quickly and monotonically from above to full CI by updating iteratively the primary, external and secondary states describing the respective static, dynamic and again static components of correlation, even when starting with a rather poor description of a strongly correlated system. In short, the iterative CI (iCI) is a very effective means toward highly correlated wave functions and ultimately full CI. ∗ To

whom correspondence should be addressed National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, State Key Laboratory of Rare Earth Materials Chemistry and Applications, College of Chemistry and Molecular Engineering, and Center for Computational Science and Engineering, Peking University, Beijing 100871, People’s Republic of China ‡ Chemistry Department, University of North Dakota, Grand Forks, ND 58202-9024, U.S.A. † Beijing

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Introduction

A strongly correlated system of electrons features generally multiple unpaired electrons or nearly degenerate electronic states, which render the single determinant description inadequate. The common paradigm for treating such systems is to decompose the overall correlation into longrange static/nondynamic (over regions comparable to the size of the orbital being correlated) and short-range dynamic (instantaneous Coulomb repulsion, mainly between opposite-spin electrons) components and treat them differently (N.B. Here, we do not distinguish the subtle difference 1 between static (due to spin/spatial symmetry) and nondynamic (due to near degeneracy) correlations, for they are generally mixed anyway). In this spirit, a vast number of multireference methods have been developed in the last 40+ years, which differ from each other in the choice of orbitals, reference space, zeroth-order Hamiltonian, and “perturbers” (i.e., the zeroth-order states allowed to interact directly with a reference state), as well as the extent to which the size-extensivity and sizeconsistency 2–4 are satisfied and the intruder problem 5,6 is avoided. A detailed comparison of all the available methods goes beyond the scope of the present work. Instead, a coarse categorization is made here, according to when the static and dynamic correlations are treated. In this context, the methods can grossly be classified into three families: (a) “static-then-dynamic” Here, the Hamiltonian projected onto a prechosen reference space VM = {Φµ ; µ = 1, · · · , dR } (0) (0) is first diagonalized to give a reference state Ψk = ∑dµR Φµ C¯µk (which may be an excited

state). This diagonalization step describes static correlation beyond the Hartree-Fock (HF) ap(0)

proximation, especially when Ψk is only qualitatively or semi-quantitatively correct. There (0)

then exists a large number of ways to determine the external (dynamic) correction to Ψk even (1)

just to first order, i.e., Ψk . The so-obtained state-specific or state-selective multireference second-order perturbation theories (SS-MRPT2) can also be characterized as “diagonalizethen-perturb” and differ 7,8 mainly in the choice of zeroth-order Hamiltonian and perturbers (0)

that are allowed to interact directly with the reference state Ψk (N.B. Although the acronym


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SS-MRPT2 is often used for a specific MRPT2, it is used here to refer more generally to any state-specific MRPT2). For instance, the perturbers can be either linear combinations 9–35 of (0)

the singly and doubly excited determinants relative to Ψk

or simply the individual excited

determinants themselves. 36–45 It is clear that the performance of such SS-MRPT2, even of the ic-MRCC (internally contracted multireference coupled-cluster) anologs, 46–54 depends criti(0)

cally on the quality of the reference state Ψk . It is often the case that a complete active space self-consistent field (CASSCF) calculation 55 or similar procedure fails to produce a good guess (0)

(0)

of Ψk . Such failure can be cured to some extent by allowing several reference states {Ψk } to remix through multi-state MRPT2 (MS-MRPT2), 56–62 especially their extended versions 63,64 (0)

that employ a zeroth-order Hamiltonian off-diagonal in the model space spanned by {Ψk }. Although such MS-MRPT2 can be characterized as “diagonalize-perturb-diagonalize”, they should still be assigned to “static-then-dynamic” instead of “static-dynamic-static” (vide post), because the second diagonalization of the effective Hamiltonian in the very small model space (0)

(0)

does not give rise to sufficient revisions of the coefficients C¯µk of any Ψk . The so-called externally corrected single-reference coupled-cluster (SRCC) approaches 65–67 can also be assigned to this family, since the dynamic exponential part of the wave function is determined in the presence of a fixed configuration interaction (CI) part that is first determined for static correlation. (b) “dynamic-then-static” Here, an effective Hamiltonian 68,69 is constructed in the whole reference space VM and is (0)

diagonalized to produce a wave function that can be (very) different from Ψk obtained by diagonalizing the bare Hamiltonian in the same space. It looks like that dynamic correction to individual model functions Φµ is considered first (without knowledge of the coefficients (0) C¯µk ) before static correlation is obtained by diagonalization. A number of SS-MRPT2 70–74

as well as MS-MRPT2 75,76 were developed in the past based on the Löwdin partitioning 77 and Bloch equation, 78 respectively. Analogous MRCC formalisms 69,79–86 were also developed in this spirit. Although very elegant, such effective Hamiltonian based approaches are 3 ACS Paragon Plus Environment


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generally plagued by the intruder problem, albeit mitigated to some extent by using an incomplete active space or general model space. 84–86 At variance with such “dynamic-then-static” approaches that all use a common set of orthonormal orbitals to construct all determinants, a different “dynamic-then-static” approach, i.e., nonorthogonal state interaction (NOSI), was also proposed, 87 where dynamic correlation (at any level of methodology) is first built into each individually optimized non-Aufbau model functions Φµ before the diagonalization in the space of nonorthogonal, dynamically correlated states is performed for static correlation. The simplest realization of NOSI was achieved by Yost et al. 88 under a different name though. It can also be combined 89 with the spin-flip nonorthogonal CI approach 90 that only accesses static correlation. All these “dynamic-then-static” approaches can also be characterized as “perturb-then-diagonalize”. (c) “static-dynamic-static” The clear distinction between the above “static-then-dynamic” and “dynamic-then-static” types of approaches lies in that the former employs the fixed (or insufficiently relaxed) coefficients (0) (0) C¯µk of the reference states Ψk , whereas the latter does not use them at all. In contrast, the

“static-dynamic-static” (SDS) type of approaches also makes use of the predetermined coeffi(0) cients C¯µk in generating external states for dynamic correlation but allow them to sufficiently

or even fully relax in the second static-correlation step. That is, the final coefficients Cµk feel the effect of both dynamic and static correlations. The intermediate Hamiltonian 91,92 based SS-MRCC, 93–103 MS-MRCC, 104–110 SS-MRPT2 111–123 and GVVPT2, 124–126 as well as MRCI (multireference configuration interaction), 127–137 all have this feature. At variance with such (unrestricted) SDS approaches that involve much more numerous secondary/buffer (NS ) (0)

and external (NQ ) states than the reference/primary states {Ψk ; k = 1, · · · , NP }, a restricted SDS framework has recently been proposed, 138 which works with the same dimension for all the three subspaces (i.e., NP = NS = NQ ). That is, the secular equation to be diagonalized is of dimension 3NP , irrespective of the numbers of correlated electrons and orbitals. Although very restricted in nature, it turns out 138 that even the lowest-order realization (i.e., SDS-MS4 ACS Paragon Plus Environment

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MRPT2, with the NP external states determined to first order without solving linear systems of equations and a zeroth-order Hamiltonian H0 for the matrix elements among themselves) is already very accurate for classic test systems of variable degeneracies. While further development of SDS-MS-MRPT2 is still of interest, we consider here a high-order realization of this restricted SDS framework through an iterative CI (iCI) scheme, which updates iteratively the primary, external and secondary states responsible for the static, dynamic and again static components of correlation, respectively. It will be shown both theoretically and numerically that iCI can converge quickly and monotonically from above to full CI (FCI). The remainder of the paper is organized as follows. In Sec. 2.1, the restricted “static-dynamicstatic” framework for strongly correlated electrons is first introduced in a heuristic manner and compared with other related approaches. The iCI approach is then presented in detail in Sec. 2.2 and compared with the ICIGSD (iterative configuration interaction general singles and doubles) method. 139 To demonstrate the efficacy of iCI, some proof-of-principle calculations are discussed in Sec. 3. Concluding remarks and possible strategies for improving the efficiency of iCI are finally given in Sec. 4.

2 2.1

Theory The restricted SDS framework

The FCI spectrum of the Schrödinger equation for a system of n electrons can generally be classified into three portions, i.e., NP low-lying, NS intermediate and NQ high-lying states. The specific values of NP , NS and NQ are of course case dependent. If NP is just one and the state is well separated from the rest, the system is only weakly correlated and can well be described by a singlereference prescription. On the other hand, if NP is exceedingly large (i.e., NP ' NP + NS + NQ ), the system would be ultra-strongly correlated and cannot be handled by standard approaches. What is particularly interesting is the intermediate case, i.e., the commonly called strongly correlated systems, with NP being a limited number. Stimulated by this general feature, we write a FCI solution 5 ACS Paragon Plus Environment


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|ΨI i (I ≤ NP ) as M

|ΨI i =

N

|Φµ iCµI ,

∑

CµI =

N = rank(C)

(1)

k=1

µ=1 N

=

∑ C¯µkC˜kI , M

∑ |Φ˜ k iC˜kI ,

˜ ki = |Φ

k=1

∑ |Φµ iC¯µk .

(2)

µ=1

N˜

≈

∑ |Φ˜ k iC˜kI .

(3)

k=1

Here, {Φµ ; µ = 1, · · · , M} represent all possible n-electron determinants (det) or equivalently con˜ k ; k = 1, · · · , N ≤ M} are to be called contracted n-electron figuration state functions (csf), while {Φ functions or simply “states”. The second equation in Eq. (1) is nothing but a singular value de¯ R† = C¯C, ˜ such that nothing has really happened when composition of the CI vector C, i.e., C = Cσ going from Eqs. (1) to (2). However, the latter implies that it is well possible to choose just a ˜ k ; k = 1, · · · , N} ˜ to represent accurately the desired state small number of contracted functions {Φ ΨI , thereby leading to Eq. (3) (N.B. If C¯ = C and hence C˜ = I, N˜ = 1 would be exact!). Stated differently, when the coefficients C˜kI are arranged in descending order in magnitude, the first N˜ ˜ k in Eq. (2) can be regarded as most important. In contrast, when workcontracted functions Φ ing with the primitive det/csf, the selection of a small number of them is usually impossible for they are not well structured in energy. In fact, as an approximation to Eq. (2), Eq. (3) has three degrees of freedom to bring into play: (a) what det/csf Φµ are to be used; (b) how to determine ˜ k are to be used. As for (c), the restricted their contraction coefficients C¯µk ; (c) how many states Φ SDS Ansatz 138 assumes simply N˜ = 3NP , that is, the same number (NP ) of primary, secondary, ˜ k in Eq. (3). The procedure goes as follows. and external states is to be employed for Φ As usually done, a reference space VM = {Φµ ; µ = 1, · · · , dR } is first constructed (following, e.g., the iterative procedure 37,38 ), which need not be CAS 55 nor that used to determine (0)

the one-particle orbitals. The lowest NP solutions of the Hamiltonian in this space, i.e., Ψk = (0) dR Φµ C¯µk , which provide either semi-quantitatively or qualitatively correct descriptions of the ∑µ=1

NP exact states, are to be taken as the primary states. For convenience, we introduce the following


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projectors dR

P =

∑ |Φµ ihΦµ | = Pm + Ps,

(4)

µ=1 NP

Pm =

(0)

∑ |Ψk

(0)

ihΨk |,

(5)

k=1

Ps = P − Pm = (1 − Pm )P,

(6)

where Pm and Ps characterize the primary (Vm ) and secondary (Vs ) parts of the reference space VM , (0)

respectively. As known from many-body perturbation theory, the first-order corrections to Ψk , viz., 1

(1)

|Ξk i = Q

(0)

(0) Ek − H0

QH|Ψk i =

(1)

∑ |ΦqiC¯qk ,

(7)

q∈Q

Q = 1 − P,

(8)

are very effective at describing dynamic correlation and hence good candidates for the second ˜ k in Eq. (3). To account for changes in the static correlation (described (external) NP functions Φ (0)

(1)

by Ψk ) due to the inclusion of dynamic correlation (described by Ξk ), the following not-energybiased Lanczos type of functions (2) |Θk i

=

(1) Ps H|Ξk i =

dR

(2)

∑ |Φµ iC¯µk

(9)

µ=1

˜ k in Eq. (3). That the cocan be introduced 138 to mimic the third (secondary) NP functions Φ (0)

(1)

efficients of Ψk , Ψk

(2)

and Θk

decrease (see Table 2 of Ref. 138 for an example) supports the

assignment of them to the corresponding functions in Eq. (3). Therefore, as a first attempt, Eq. (3) reads (1)

|ΨI i =

NP

(0)

(1)

NP

(1)

(1)

NP

(2)

(1)

∑ |Ψk iC˜kI + ∑ |Ξk iC˜(k+NP)I + ∑ |Θk iC˜(k+2NP)I .

k=1

k=1

k=1


(10)


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The yet unknown coefficients C˜ (1) are to be determined by the generalized secular equation

HC˜ (1) = SC˜ (1) E.

(11)

Several remarks can be made here. (1)

(1) It can readily be seen from Eqs. (7) and (9) that both the external state Ξk and secondary state (2)

(0)

Θk are fully contracted and specific to each primary state Ψk . Therefore, the dimension of Eq. (11) is only 3NP , irrespective of the numbers of correlated electrons and orbitals. This Ansatz can hence be termed “minimal MRCI”, more precisely ixc-MRCISD+s (internally and externally contracted MRCI with singles and doubles and augmented with secondary states). (1)

are correct up to first order, not to be confused by the superscript.

(2)

relative to Ψk

(2) The wave functions ΨI

The formal order of Θk

(0)

is not really meaningful for they are used here as (0)

independent functions. Likewise, the second NP solutions {Ψk+NP ; k = 1, · · · , NP } of the pro(2)

jected Hamiltonian PHP can also be used in place of {Θk } for the purpose of “buffering”, (0)

(1)

in the sense that they interact with Ψk only indirectly through the interaction with Ξk , viz. (0)

(0)

(2)

(0)

(0)

(1)

(2)

(1)

hΨl |Ps HPm |Ψk i = hΘl |Ps HPm |Ψk i = 0, hΨl |Ps HQ|Ξk i 6= 0, and hΘl |Ps HQ|Ξk i 6= 0, ∀l, k. That is, the Hamiltonian matrix in Eq. (11) is of the following structure 



0  Pm HPm Pm HQ    H= QHP QHQ QHP m s     0 Ps HQ Ps HPs

 or 3NP ×3NP

0 Pm HPm   0 Ps HPs   QHPm QHPs

 Pm HQ  Ps HQ    QHQ

(12) 3NP ×3NP

It has been shown 138 that the two choices of secondary states give rise to similar numerical (2)

results but the {Θk } ones are preferred, for they see directly the dynamic correlation, cf. Eq. (9). (3) Although only a first-order termination of the contraction coefficients for the external states has been considered above, any other state-specific approach for dynamic correlation can also 8 ACS Paragon Plus Environment

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be adopted. Therefore, the restricted SDS framework is only restricted in the structure (10) of the wave function and has sufficient degrees of freedom for improvement. 138 (4) No matter how the external space is contracted, to the lowest or a higher order, the ixcMRCISD+s wave function (10) is linearly variational, which facilitates subsequent evaluations of energy gradients and properties. (5) The ixc-MRCISD+s approach can best be viewed as a particular representation of the bare Hamiltonian in a specially designed n-electron basis. (6) As a truncated MRCI approach, ixc-MRCISD+s is neither size-extensive nor size-consistent. However, the situation can readily be improved by using matrix-dressing techniques 140,141 or coupled-cluster type of corrections. 142–145 At this stage it is instructive to compare ixc-MRCISD+s with other related approaches. (i) If the reference space VM is not contracted (i.e., Pm = P and Ps = 0), both doubly contracted (dc) 146 and externally contracted (xc) MRCISD 147,148 can be obtained by confining properly the summation over the compound index q in Eq. (7). Still, however, there exists a subtle but important difference between them and the present state-specific form (7). Specifically, the first-order contraction coefficients in both dc-MRCISD and xc-MRCISD are determined by using only one reference state, i.e., (1) C¯qk =

hq|H|0i , E0 − hq|H0 |qi

∀k,

(13)

(0)

where |0i of energy E0 is the energetically lowest one among the zeroth-order states {Ψl }. (1)

On the other hand, if only the {Ξk } are replaced with uncontracted external singles and contracted external doubles (with coefficients C¯ (0) ), Eq. (10) would correspond to the internally contracted MRCISD. 131–134 (ii) The ixc-MRCISD+s approach is also related to MS-MRPT2 56–64 but only in structure. This 9 ACS Paragon Plus Environment


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(1) (1) (1) can be seen by setting C˜kI = C˜(k+NP )I and C˜(k+2NP )I = 0 in Eq. (10), leading to

(1) ΨI

NP

=

(0)

∑ (Ψk

(1)

(1)

+ Ξk )C˜kI .

(14)

k=1

The Hamiltonian matrix (11) correct to second order is then of dimension NP , viz., [2]

(He f f )kl

1 (0) (1) (0) (0) (0) (1) {hΨk + Ξk |H|Ψl i + hΨk |H|Ψl + Ξl i} 2 1 (0) (0) (1) (0) (0) (1) = hΨk |H|Ψl i + {hΞk |H|Ψl i + hΨk |H|Ξl i}. 2

=

(15) (16)

It is well known that such multi-state approaches do not have full flexibility for the revision of the components of the wave function in the primary space and should therefore be assigned to the “static-then-dynamic” instead of “static-dynamic-static” family of methods. Moreover, they suffer from the (in)famous intruder state problem. Both situations can be improved by (2)

further introducing the secondary states {Θk }, i.e., NP

NP

k

k

˜ (1) = ∑(Ψ(0) + Ξ(1) )C˜ (1) + ∑ Θ(2)C˜ (1) Ψ , I k k kI k (k+NP )I

(17)

which leads to a second-order intermediate Hamiltonian 91,92 of dimension 2NP : [2] [2] (H˜ e f f )kl = (He f f )kl ,

k, l = 1, · · · , NP ,

[2] (2) (1) (H˜ e f f )(k+NP )l = hΘk |H|Ξl i, [2]

(2)

(1) (2) (H˜ e f f )k(l+NP ) = hΞk |H|Θl i,

(2)

(H˜ e f f )(k+NP )(l+NP ) = hΘk |H|Θl i.

(18) (19) (20)

At variance with such a restricted form, a practical intermediate Hamiltonian approach (e.g., (0)

GVVPT2 124–126 ) usually takes all the {ΨK , K = NP + 1, · · · , dR } states as the secondary states. In this case, several side effects may arise: (a) Such secondary states are usually much more numerous than the target primary states (i.e., dR 2NP ) and some of them may actually be higher in energy than some of the external states, thereby losing the meaning of static 10 ACS Paragon Plus Environment

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correlation. Moreover, treating higher secondary states better than lower external states is certainly unbalanced. (b) Probably more serious, due to the one-to-one fixed combination (0)

(1)

(0)

(1)

˜ (17) ), the relaxation of the primary states Ψ furnished by the sec(i.e., Ψk + Ξk in Ψ I k ondary states is accompanied with the reduction of dynamic correlation (i.e., the coefficient (1) (1) C˜kI of Ξk is less than 1 in magnitude), as compared with the intermediate normalization (1)

(1)

(i.e., C˜kI = 1). All such problems are avoided by treating the external states {Ξk } as independent basis functions, the key feature of ixc-MRCISD+s and its perturbation variants. 138 In sum, ixc-MRCISD+s combines the merits of both internally and externally contracted MRCI as well as intermediate Hamiltonian approaches. It deserves to be mentioned again that (0)

(1)

(2)

(1)

the {Ψk }, {Ξk } and {Θk } states do have decreasing weights in the wave functions ΨI (cf. Table 2 of Ref. 138 ), thereby justifying the “static-dynamic-static” characterization.

(iii) The three-subspace philosophy has long been used in configuration selection-extrapolation approaches 37,38 but in a different way. The aim there is to construct as small as possible a primary subspace Vm as well as a secondary (intermediate) subspace Vs , by adding in progressively individual configurations based on the magnitude of their second-order energies. Once this is done, the corrections of Vs to Vm are accounted for either variationally or at a high order of perturbation theory, whereas the corrections of the external singles and doubles to Vm are accounted for at second order. Although called “selected CI”, such approaches are intrinsically SS-MRPT2 in the spirit of intermediate Hamiltonian but belong to the “staticthen-dynamic” family of methods. It is just that the (arbitrary) reference space Vm ⊕ Vs is constructed in a more rational manner. The CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) approach 37 designed in this spirit can indeed provide very accurate results, especially when the reference space Vm ⊕ Vs already accounts for a large portion (say, ∼90%) of the total correlation and meanwhile higher-order dynamic correlation is negligible. 149–151 However, such situations are very rare in practice. Therefore, an extrapolation step is usually taken to estimate the final energy of “selected CI”. 37,38 Compared with the present variational ixc-MRCISD+s, the interaction Ps HQ between the sec11 ACS Paragon Plus Environment


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ondary/intermediate and external subspaces is missing in such approaches.

2.2

The iCI approach

Taking ixc-MRCISD+s as a start, various simplifications and extensions have been proposed. 138 For instance, when the QHQ block of the Hamiltonian matrix (12) is replaced with QH0 Q (shifted appropriately for aligning the energy scale), the so-called SDS-MS-MRPT2 approach is obtained, which scales computationally as n5 and can describe very well very difficult systems. Among the various possible extensions of ixc-MRCISD+s for even higher accuracy, we here consider an iterative CI (iCI) scheme, which updates the primary, external and secondary states iteratively until the FCI space is reached. To this end, we can simply rewrite Eq. (10) as NP

(i)

(i−1)

∑ |Ψk

|ΨI i =

k=1

NP

(i)

(i)

(i)

NP

(i+1)

iC˜kI + ∑ |Ξk iC˜(k+NP )I + ∑ |Θk k=1

k=1

(i)

iC˜(k+2NP )I ,

(21)

with i = 1, 2, · · · being the iteration number (N.B. i = 0 refers to an initial HF or MCSCF step). (i)

However, at variance with Eq. (7), the Ξk functions are here defined as (i)

1 (i−1) Q(i−1) H|Ψk i, (i−1) Ek − H0 NP (i−1) (i−1) (i−1) 1 − Pm = 1 − |Ψl ihΨl |. l=1

|Ξk i = Q(i−1) Q(i−1) =

(22) (23)

∑

The distinction between the projectors Q (8) and Q(i−1) (23) lies in that the latter gives rise to (i)

(i+1)

functions Ξk (22) that belong to both the external and secondary subspaces. Yet, the Θk

states

can still be obtained in the same way as Eq. (9), viz., (i+1)

|Θk

(i−1)

i = Ps

(i)

H|Ξk i,

(i−1)

Ps

(i−1)

= P(i−1) − Pm

,

(24) (i−1)

where P(i−1) is defined by the det/csf used to construct the current primary states Ψk (i−1)

Q(i−1) = 1 − Pm

(i−1)

− Ps

. Had

been used in Eq. (22), the external and hence the secondary subspaces


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would be null when the FCI space has been reached. One then has no chance to reach FCI accuracy (i−1)

because the contraction coefficients of Ψk

can no longer be changed. As for the resolvent in

Eq. (22), H0 is just the diagonal part of the full Hamiltonian in the determinant basis, 152,153 while (i−1)

Ek

are the eigenvalues of Eq. (11) from the preceding iteration.

The above iterative scheme does not impose any restrictions on the external subspace of a given iteration. Therefore, the external subspace gets expanded by two ranks (i.e., singles and doubles) per iteration until the FCI space is reached. Yet, some refinement of such (macro) iterations is desirable because the first-order determination (22) of the contraction coefficients of the external states is not optimal. Their relaxation can be achieved by introducing some micro-iterations j within a macro-iteration i: (i, j)

|Ξk

Q(i, j−1) = = (i, j+1)

|Θk

i =

1

(i, j−1) Q(i, j−1) H|Ψk i, (i, j−1) Ek − H0 (i, j−1) 1(i) − Pm NP (i, j−1) (i, j−1) 1(i) − |Ψl ihΨl |, l=1 (i, j−1) (i, j) (i, j−1) (i, j−1) Ps H|Ξk i, Ps = P(i, j−1) − Pm ,

i = Q(i, j−1)

∑

(25)

(26) (27)

where a superscript has been appended to the identity to emphasize its dimensionality. Computationally, this is a rather quick step, since the action of the Hamiltonian on the external states is already available when constructing the secondary states. It has been verified numerically that the minor errors due to the contractions can be removed completely by performing a sufficient number of micro-iterations, which should become more effective if a somewhat larger NP is used in Eq. (26). In other words, at the Mth macro-iteration, sufficient micro-iterations will reproduce uncontracted MRCI up to 2M-tuple excitations relative to the initial guess (e.g., MRCISDTQ for M = 2). The iCI scheme is hence fundamentally different from the iterative procedure invoked in the selected CI (more precisely SS-MRPT2) approaches 36–40 for constructing the primary and secondary subspaces. The iCI can also be characterized as an alternative scheme for diagonalizing a large matrix for 13 ACS Paragon Plus Environment


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a few eigenpairs, with the following merits: (a) Instead of diagonalizing directly the target matrix A (e.g., that of MRCISD) through matrix-vector products, the matrix A is folded, a priori, into a very small (3NP × 3NP ) matrix with physically well-defined trial vectors, yet without knowing the target matrix A itself. This is possible because the target matrix A is known to be a particular representation of a known operator in a known basis. The micro-iterations are then invoked to relax the contractions without expanding the trial vectors, which is in sharp contrast with the existing Krylov subspace iterative methods (e.g., the Davidson scheme 154 ) in which the trial vectors are substantially expanded along with iterations. Moreover, for practical purposes, such expansions need to be restarted, often with ill-defined conditions. (b) The domain and range of matrix A to be targeted do get expanded with the macro-iterations (e.g., that of MRCISDTQ at the second macro-iteration) but this is for another basis representation and a higher accuracy. (c) In view of the variational nature, both the macro- and micro-iterations are guaranteed to converge monotonically thanks to the MacDonald-Hylleraas-Undheim interlace theorem. 155,156 At this stage it is instructive to compare the present iCI (iterative Configuration Interaction) with the groundbreaking ICI (Iterative Configuration Interaction) method of Nakatsuji, 157–159 especially the ICIGSD variant: 139 (a) While the iCI is introduced here based entirely on physical intuitions and can best be viewed as a particular discretization of the first-quantized many-electron Hamiltonian in a specially designed (through contractions) and iteratively updated n-electron basis, the ICI 157,158 was stimulated from the mathematical structure of the exact wave function and based on the division of the first- or second-quantized many-electron Hamiltonian into parts, without explicit recourse to a particular n-electron basis. (b) Both the iCI and ICI 157,158 are generic. The former is a shorthand notation for SDS-ICI, which permits various treatments of dynamic correlation as discussed above, whereas the latter is a shorthand notation for ICIND, which allows for different divisions of the full Hamiltonian. Here, the abbreviation “ND” stands for “number of divisions”, which ranges from one (i.e., the whole first- or second-quantized many-electron Hamiltonian is taken as a single part) to the 14 ACS Paragon Plus Environment

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total number of variables in the second-quantized many-electron Hamiltonian (i.e., the number of general singles and doubles (GSD); see equation (2.4) in Ref. 157 ). (c) The iCI is more akin to the ICIGSD. 139 In terms of the present notation, in the ith iteration, the ICIGSD calculation for state k amounts to discretizing the second-quantized many-electron (i)

(i−1)

Hamiltonian in the space VGSD,k spanned by functions {|Ψk

(i−1)

i, a†p aq |Ψk

(i−1)

i, a†p a†q as ar |Ψk

i},

which are nonorthogonal, internally contracted but externally uncontracted. At first glance, the present iCI is merely an externally contracted version of ICIGSD, provided that the secondary (i+1)

states {Θk

} are not considered in the former. However, this is not really the case. The

ICIGSD is by construction a single-state approach, 158 that is, one ICISD calculation can only get one exact (FCI) state, with the other solutions being only approximations to the excited states. A remedy to this “excited-state problem” is to make a single-step diagonalization of (i)

the Hamiltonian in an enlarged space (by augmenting VGSD,k with higher-rank excitation operators) after the ICIGSD iterations have converged. In contrast, the present iCI does get NP exact (FCI) states simultaneously. This is because the micro-iterations of iCI allow for full relaxation of the NP states simultaneously. Another important distinction between the iCI and ICIGSD lies in that the former is compatible with explicit correlation, whereas the latter is not, due to the fact that the second-quantized many-electron Hamiltonian adopted by the latter only has a finite spectrum limited by the chosen one-particle basis. (d) Both the iCI and ICIGSD can be interpreted as special ways for partial diagonalization of the FCI matrix and share the same merits depicted above. However, the dimension of the secular equation (11) of ICIGSD is orders of magnitude larger than that of iCI. More specifically, the dimension of ICIGSD is in the order of n2o n2v , with no and nv being the numbers of occupied and virtual orbitals, respectively, which is to be compared with the dimension 3NP of iCI, with NP being typically one or two times the number of target states (independent of the numbers of correlated electrons and orbitals). Moreover, because of the introduction of micro-iterations at each macro-iteration, the present iCI should in principle converge faster than the ICIGSD in



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terms of the number of macro-iterations, although numerical examples may not be always the case. The above formal comparison between iCI and ICIGSD indicates strongly that ‘the smaller iCI is better than the larger ICI’. Whether this is true in practice remains to be confirmed by implementing both iCI and ICIGSD into the same code, so as to compare directly their relative efficiency. On the fundamental point, it is curious to ask whether the exact conditions 157,158 underlying the singlestate ICIND/ICIGSD can be generalized rigorously to multiple states.

3

Results and discussion

To assess the efficacy of the iCI approach, the well-studied symmetric stretching of water was first considered. The equilibrium geometry (Re ) has the hydrogens at (0., ±1.515263, −1.049898) a.u., with oxygen at the origin, while the other geometries were obtained by elongating simultaneously the OH bonds by 2.0, 2.5 and 3.0 times. The calculations employed the cc-pVDZ basis, 160 two primary states, and one additional micro-iteration at each macro-iteration. As the overall monotonic convergence behavior is theoretically guaranteed, the main concern here is actually the rate (0)

of convergence, which is directly related to the quality of the initial primary states |Ψk i. The two equally weighted CASSCF(4,4) (i.e., 4 electrons in 4 active orbitals) states are obviously a natural choice. For comparison, a worse starting point was also considered: The first primary state was taken to be the restricted Hartree-Fock (RHF) function |ΨRHF i, while the second was generated by orthogonalizing H|ΨRHF i against |ΨRHF i. The deviations of the so-calculated iCI energies from the FCI values 161 were plotted in Fig. 1. The monotonic convergence is obvious for both sets of initial primary states. It is just that the rate of convergence is somewhat different. More specifically, at Re , both RHF-iCI and CASSCF-iCI converge to micro-Hartree (µH) accuracy within 4 macro-iterations (N.B. The RHF-ICIGSD converges within two iterations but with a minimal basis and the O1s orbital frozen, see Table 3 of Ref. 139 ). However, at 2.0Re , RHF-iCI requires 10 macroiterations, twice those by CASSCF-iCI. Anyhow, the former represents a strong test of the ability 16 ACS Paragon Plus Environment

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of iCI to recover not only dynamic but also static correlations. Moreover, it is worthy of noting that CASSCF-iCI converges somewhat faster at 3.0Re than at 2.5Re and even 2.0Re . However, this is hardly surprising for the quality of the underlying CASSCF is somewhat better in the bond broken (3.0Re ) than in the bond breaking (2.0Re ∼ 2.5Re) regions. Although not yet implemented, the convergence can certainly be improved by using natural orbitals derived from the correlated wave function at an early macro-iteration. The dependence on the initial primary states is then removed thereafter. To better appreciate the rate of convergence of CASSCF-iCI, a brief comparison with the ec-CCSDt-CASSCF (externally corrected coupled-cluster with singles, doubles and active triples employing four- and five-body clusters from CASSCF) and CCSDT approaches 67 can be made here. While ec-CCSDt-CASSCF has submillihartree errors for all but the pernicious 2.5Re geometries, CCSDT has errors of tens of mH at the stretched geometries. In contrast, for all the considered geometries, the errors of CASSCF-iCI are only 100 and 10 µH after just 2 and 3 macro-iterations, respectively. To reveal the role played by the secondary states, the same calculations but without such states were also carried out, with the results plotted in Fig. 2. It is seen that excluding the secondary states worsens the convergence by one or two iterations for both sets of initial primary states. Of course, the iteration count is not the only measure of efficiency for an iteration without the secondary states is quicker than that with them. At this stage one may argue that the cc-pVDZ basis set is a bit too small for correlation, such that the above findings may not hold for larger basis sets. To address this issue, we performed iCI calculations on equilibrium water with the cc-pVTZ basis, 160 one primary state, and two micro-iterations at each macro-iteration. Since the calculations were rather expensive and the corresponding FCI results were not available for comparison, only results of the first two macroiterations were reported in Table 1, to be compared with the MRCISD, MRCISD(TQ) 162 and ec-CCSDt-CASSCF 67 results. Several observations can readily be made. First, one iteration of iCI, especially with one micro-iteration, produces results that are in very good agreement with uncontracted MRCISD (i.e., 0.2 mH for cc-pVDZ and 2.2 mH for cc-pVTZ). Since a micro-iteration



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of iCI is computationally similar with one iteration of MRCISD, this result in itself is notable. Second, two iterations of iCI, again with 1 or 2 additional micro-iterations, produce results that are better than those by both MRCISD(TQ) and ec-CCSDt-CASSCF. The smooth monotonic convergence behavior also allows us to extrapolate the iCI energy as function of the iteration number (∗)

(i+1)

by using, e.g., Ek = Ek

(i+1)

− (Ek

(i)

(i+1)

− Ek )2 /(Ek

(i)

(i−1)

− 2Ek + Ek

). In the case of cc-pVDZ,

the so-extrapolated FCI deviates from the true FCI by 88 µH, indicating that such an estimate of FCI is truly meaningful. It can then be reckoned that the cc-pVTZ iCI with two macro-iterations and two micro-iterations is about 0.5 mH away from the yet unknown cc-pVTZ FCI. It is also interesting to see that the present iCI result is marginally lower and hence somewhat better than the most recent DMRG (density matrix renormalization group) value. 67 It can hence be concluded that iCI converges quickly to submillihartree accuracy for chemically useful basis sets. This is a very significant finding, since exceedingly large basis sets would not be needed when iCI is combined with the idea of explicit correlation. To assess the performance of iCI for excited states, the first two 1 A1 states of equilibrium water were calculated using the cc-pVDZ basis. Except for the lowest, equally averaged two CASS(0)

CF(4,4) states |Ψk i (k = 1, 2), additional two primary states were generated by orthogonalizing (0)

(0)

H|Ψk i against |Ψk i. Such states are presumably better than the lowest, equally averaged four states of CASSCF(4,4). One additional micro-iteration was performed at each macro-iteration (without secondary states). After just two macro-iterations, the results (i.e., -76.241744 a.u. for 11 A1 and -75.860361 a.u. for 21 A1 ) are already very accurate, viz., 0.1 mH from the corresponding uncontracted MRCISDTQ56 results (-76.241860 a.u. for 11 A1 and -75.860486 a.u. for 21 A1 ). The accuracy was further enhanced to a few µH by the third macro-iteration (i.e., -76.241858 a.u. for 11 A1 and -75.860478 a.u. for 21 A1 ). Even such small errors can be removed by further micro-iterations for three macro-iterations of iCI access the same space as MRCISDTQ56. Finally, we mention that similar calculations on LiF were also carried out. The convergence was achieved rather quickly around the ionic-covalent avoided-crossing region even when starting with RHF. That is, compared with water, no new information on the convergence pattern of iCI


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was noticed. Therefore, such results are not presented here.

4

Conclusions and outlook

A specific realization of the “static-dynamic-static” framework 138 beyond the second-order level (i.e., SDS-MS-MRPT2) has been achieved through the iCI approach, which has the full capability to describe both the static and dynamic components of correlation, as well as their mutual influences, in a dynamically balanced manner. This is made possible by updating iteratively the primary, external and secondary states responsible for the static, dynamic and again static components of correlation, respectively. Each iteration accesses a space that is higher by two ranks than that of the preceding iteration. That is, up to 2M-tuple excitations relative to the initial primary space can be accessed if M iterations are carried out. Due to the variational nature, any minor loss of accuracy stemming from the contractions can be reduced/removed by carrying out some microiterations. In other words, by controlling the numbers of macro- and micro-iterations, the iCI will generate a series of contracted/uncontracted single/multirefeence CISD· · · 2M, with the resulting energy being physically meaningful at each level. Numerical results for the oft used symmetric stretching of H2 O showed that the CASSCF-iCI approach does approach FCI (to within µH) within a small number of iterations (typically 4 ∼ 6), with the higher number associated with regions of bond breaking and commensurate decrease in the quality of the CASSCF reference. The CASSCF-iCI results after just two iterations are already better than those of operationally more involved approaches. The situation is only slightly worse when a poor reference state, such as RHF, is taken as the start. This should be the case even if the starting reference state has little weight in the final wave function, unless it has strictly zero weight due to symmetry reasons (nobody would do this!). Another appealing feature of iCI lies in that it can describe low-lying excited states with the same procedure and accuracy as the ground state. Although more extensive tests are necessary, the present findings should always hold thanks to the underlying variational nature.



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Although computational efficiency was not a prime consideration in developing the pilot computer program, it should be noted that the cc-pVTZ calculations on water, which entailed 8 × 108 determinants, could be performed on a single processor of a commercially available workstation. Like any other methods, the efficiency of iCI can only be gained by truncating appropriately the external orbital and configuration spaces. For the former, the so-called improved virtual orbitals, 163–165 frozen natural orbitals 166 or pair natural orbitals 167–170 can be used. Among the various selection schemes for truncating the external configuration space, the direct general CI 171 for arbitrarily selected spin-complete orbital configurations (i.e., orbital configurations coupled in all possible ways with the spin functions) and the a priori selection scheme 172 are most appealing. In the simplest realization, external excitations from those configurations with very small coefficients in the primary states can be excluded. This can be facilitated by using an adaptive macroconfiguration description of electron distributions. When generating the secondary states, a selection of the external configurations can also be invoked based on their first-order coefficients. Both selections should benefit from a local representation. Since M iterations of iCI already reproduce uncontracted MRCI up to 2M-tuple excitations (relative to the initial guess), the active space can gradually be reduced in subsequent iterations, that is, the effects of still higher-rank excitations are to be accounted for only within a limited space. The secondary states, which play some role only in the beginning of the iterations, can also be discarded. Moreover, instead of the first-order determination of the external states, any other state-specific, internally-contracted multireference approach (e.g., multiconfiguration coupled electron pair approximation 94 ) may be used. In particular, dressed CI 140,141 or an energy functional 142–145 that is approximately size-extensive can also be used in place of the straight eigenvalue problem (11). It can therefore be envisaged that the iCI approach can be made to achieve very high accuracy yet with reasonable computational effort.


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Acknowledgement The research of this work was supported by NSFC (Project Nos. 21273011, and 21290192) and NSF (Grant No. IIA-1355466).



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References (1) Lyakh, D. I.; Musiaz, M.; Lotrich, V. F.; Bartlett, R. J. Chem. Rev. 2012, 112, 182–243. (2) Pople, J.; Binkley, J.; Seeger, R. Int. J. Quantum Chem. 1976, 10 (S10), 1–19. (3) Bartlett, R. J.; Purvis, G. D. Phys. Scr. 1980, 21, 255–265. (4) Bartlett, R. J. Annu. Rev. Phys. Chem. 1981, 32, 359–401. (5) Schucan, T. H.; Weidenmüller, H. A. Ann. Phys. 1972, 73, 108–135. (6) Schucan, T. H.; Weidenmüller, H. A. Ann. Phys. 1973, 76, 483–509. (7) Kozlowski, P. M.; Davidson, E. R. J. Chem. Phys. 1994, 100, 3672–3682. (8) Malrieu, J. P.; Heully, J. L.; Zaitsevskii, A. Theor. Chim. Acta 1995, 90, 167–187. (9) Wolinski, K.; Sellers, H.; Pulay, P. Chem. Phys. Lett. 1987, 140, 225–231. (10) Kucharski, S.; Bartlett, R. Int. J. Quantum Chem. 1988, 34 (S22), 383–405. (11) McDouall, J. J. W.; Peasley, K.; Robb, M. A. Chem. Phys. Lett. 1988, 148, 183–189. (12) Wolinski, K.; Pulay, P. J. chem. Phys. 1989, 90, 3647–3659. (13) Murphy, R. B.; Messmer, R. P. Chem. Phys. Lett. 1991, 183, 443–448. (14) Murphy, R. B.; Messmer, R. P. J. Chem. Phys. 1992, 97, 4170–4184. (15) Zarrabian, S.; Paldus, J. Int. J. Quantum Chem. 1990, 38, 761–778. (16) Kozlowski, P. M.; Davidson, E. R. Chem. Phys. Lett. 1994, 222, 615–620. (17) Davidson, E. R. Chem. Phys. Lett. 1995, 241, 432–437. (18) Pariser, O.; Ellinger, Y. Chem. Phys. 1996, 205, 323–349.


Page 22 of 35

Page 23 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(19) Andersson, K.; Malmqvist, P.-A.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. J. Phys. Chem. 1990, 94, 5483–5488. (20) Andersson, K.; Malmqvist, P.-A.; Roos, B. O. J. Chem. Phys. 1992, 96, 1218–1226. (21) Werner, H.-J. Mol. Phys. 1996, 89, 645–661. (22) Celani, P.; Werner, H.-J. J. Chem. Phys. 2000, 112, 5546–5557. (23) van Dam, H. J. J.; Lenthe, J. H. V. Mol. Phys. 1998, 93, 431–439. (24) van Dam, H. J. J.; Lenthe, J. H. V.; Ruttink, P. J. A. Int. J. Quantum Chem. 1999, 72, 549– 558. (25) Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. J. Chem. Phys. 2001, 114, 10252–10264. (26) Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. Chem. Phys. Lett. 2001, 350, 297–305. (27) Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. J. Chem. Phys. 2002, 117, 9138–9153. (28) Angeli, C.; Borini, S.; Cimiraglia, R. Theor. Chem. Acc. 2004, 111, 352–357. (29) Rolik, Z.; Szabados, A.; Surján, P. R. J. Chem. Phys. 2003, 119, 1922–1928. (30) Rolik, Z.; Szabados, A.; Surján, P. R. J. Chem. Phys. 2005, 122, 114104. (31) Zoboki, T.; Szabados, A.; Surján, P. R. J. Chem. Theory Comput. 2013, 9, 2602–2608. (32) Jeszenszki, P.; Nag, P. R.; Zoboki, T.; Szabados, A.; Surján, P. R. Int. J. Quantum Chem. 2014, 114, 1048–1052. (33) Xu, E.; Li, S. J. Chem. Phys. 2013, 139, 174111. (34) Chen, Z.; Song, J.; Shaik, S.; Hiberty, P. C.; Wu, W. J. Phys. Chem. A 2009, 113, 11560– 11569. 23 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(35) Chen, Z.; Chen, X.; Ying, F.; Gu, J.; Zhang, H.; Wu, W. J. Chem. Phys. 2014, 141, 134118. (36) Huron, B.; Malrieu, J. P.; Rancurel, P. J. Chem. Phys. 1973, 58, 5745–5759. (37) Evangelisti, S.; Daudey, J. P.; Malrieu, J. P. Chem. Phys. 1983, 75, 91–102. (38) Bruna, P. J.; Peyerimhoff, S. D.; Buenker, R. J. Chem. Phys. Lett. 1980, 72, 278–284. (39) Buenker, R. J.; Peyerimhoff, S. D. Theor. Chem. Acta 1974, 35, 33–58. (40) Buenker, R. J.; Peyerimhoff, S. D. Theor. Chem. Acta 1975, 39, 217–228. (41) Hirao, K. Chem. Phys. Lett. 1992, 190, 374–380. (42) Hirao, K. Chem. Phys. Lett. 1992, 196, 397–403. (43) Hirao, K. Int. J. Quantum Chem. 1992, 44 (S26), 517–526. (44) Hirao, K. Chem. Phys. Lett. 1993, 201, 59–66. (45) Chen, F.; Fan, Z. J. Comput. Chem. 2014, 35, 121–129. (46) MahaPatra, U. S.; Datta, B.; Bandyopadhyay, B.; Mukherjee, D. Adv. Quantum Chem. 1998, 30, 163–193. (47) Evangelista, F. A.; Gauss, J. J. Chem. Phys. 2011, 134, 114102. (48) Hanauer, M.; Köhn, A. J. Chem. Phys. 2011, 134, 204111. (49) Hanauer, M.; Köhn, A. J. Chem. Phys. 2012, 137, 131103. (50) Sinha, D.; Maitra, R.; Mukherjee, D. Comput. Theor. Chem. 2013, 1003, 62–70. (51) Yanai, T.; Chan, G. K.-L. J. Chem. Phys. 2006, 124, 194106. (52) Neuscamman, E.; Yanai, T.; Chan, G. K.-L. J. Chem. Phys. 2010, 132, 024106. (53) Li, S. J. Chem. Phys. 2004, 120, 5017–5026. 24 ACS Paragon Plus Environment

Page 24 of 35

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(54) Fang, T.; Li, S. J. Chem. Phys. 2007, 127, 204108. (55) Roos, B. O.; Taylor, P. R.; Siegbahn, P. E. M. Chem. Phys. 1980, 48, 157–173. (56) Spiegelmann, F.; Malrieu, J.-P. J. Phys. B: At. Mol. Opt. Phys. 1984, 17, 1235–1257. (57) Nakano, H. J. Chem. Phys. 1993, 99, 7983–7992. (58) Nakano, H. Chem. Phys. Lett. 1993, 207, 372–378. (59) Ebisuzaki, R.; Watanabe, Y.; Nakano, H. J. Comput. Chem. 2002, 23, 1166–1175. (60) Ebisuzaki, R.; Watanabe, Y.; Nakano, H. Chem. Phys. Lett. 2007, 442, 164–169. (61) Finley, J.; Malmqvist, P.-A.; Roos, B. O.; Serrano-Andrés, L. Chem. Phys. Lett. 1998, 288, 299–306. (62) Angeli, C.; Borini, S.; Cestari, M.; Cimiraglia, R. J. Chem. Phys. 2004, 121, 4043–4049. (63) Granovsky, A. A. J. Chem. Phys. 2011, 134, 214113. (64) Shiozaki, T.; Gy˝orffy, W.; Celani, P.; Werner, H.-J. J. Chem. Phys. 2011, 135, 081106. (65) Li, X.; Paldus, J. J. Chem. Phys. 1997, 107, 6257–6282. (66) Peris, G.; Planelles, J.; Malrieu, J.-P.; Paldus, J. J. Chem. Phys. 1999, 110, 11708–11716. (67) Xu, E.; Li, S. J. Chem. Phys. 2015, 142, 094119. (68) Lindgren, I. Int. J. Quantum Chem. 1978, 14 (S12), 33–58. (69) Durand, P. Phys. Rev. A 1983, 28, 3184–3192. (70) Gershgorn, Z.; Shavitt, I. Int. J. Quantum Chem. 1968, 2, 751–759. (71) Davidson, E. R.; Murchie, L. E. M.; Day, S. J. J. Chem. Phys. 1981, 74, 5491–5496. (72) Shavitt, I. Chem. Phys. Lett. 1992, 192, 135–137. 25 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(73) Rawlings, D. C.; Davidson, E. R. Chem. Phys. Lett. 1983, 98, 424–427. (74) Lei, Y.; Wang, Y.; Han, H.; Song, Q.; Suo, B.; Wen, Z. J. Chem. Phys. 2012, 137, 144102. (75) Brandow, B. H. Rev. Mod. Phys. 1967, 39, 771–828. (76) Lindgren, I. J. Phys. B At. Mol. Opt. Phys. 1974, 7, 2441–2470. (77) Löwdin, P. O. J. Chem. Phys. 1951, 19, 1396–1401. (78) Bloch, C.; Horowitz, J. Nucl. Phys. 1958, 8, 91–105. (79) Mukherjee, D.; Moitra, R. K.; Mukhopadhyay, A. Mol. Phys. 1977, 33, 955–969. (80) Lindgren, I.; Mukherjee, D. Phys. Rep. 1987, 151, 93–127. (81) Haque, A.; Kaldor, U. Chem. Phys. Lett. 1985, 117, 347–351. (82) Jeziorski, B.; Paldus, J. J. Chem. Phys. 1989, 90, 2714–2731. (83) Jeziorski, B.; Monkhorst, H. Phys. Rev. A 1981, 24, 1668–1681. (84) Mukherjee, D. Chem. Phys. Lett. 1986, 125, 207–212. (85) Meissner, L.; Kucharski, S. A.; Bartlett, R. J. J. Chem. Phys. 1989, 91, 6187–6194. (86) Li, X.; Paldus, J. J. Chem. Phys. 2003, 119, 5320–5333. (87) Liu, W. Mol. Phys. 2010, 108, 1679–1706. (88) Yost, S. R.; Kowalczyk, T.; Voorhis, T. V. J. Chem. Phys. 2013, 139, 174104. (89) Huang, C.; Liu, W. (unpublished) (90) Mayhall, N. J.; Horn, P. R.; Sundstrom, E. J.; Head-Gordon, M. Phys. Chem. Chem. Phys. 2014, 16, 22694–22705. (91) Kirtman, B. J. Chem. Phys. 1981, 75, 798–808. 26 ACS Paragon Plus Environment

Page 26 of 35

Page 27 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(92) Malrieu, J. P.; Durand, P. H.; Daudey, J. P. J. Phys. A: Math. Gen. 1985, 18, 809–826. (93) Ruttink, P. J. A.; van Lenthe, J. H.; Zwaans, R.; Groenenboom, G. C. J. chem. Phys. 1991, 94, 7212–7220. (94) Fink, R.; Staemmler, V. Theor. Chim. Acta 1993, 87, 129–145. (95) Huba˘c, I.; Neogrády, P. Phys. Rev. A 1994, 50, 4558–4564. (96) Masik, J.; P., I. H.; Mach, J. Chem. Phys. 1998, 108, 6571–6579. (97) MahaPatra, U. S.; B. Datta, B. B.; Mukherjee, D. Mol. Phys. 1998, 94, 157–171. (98) Mahapatra, U. S.; Datta, B.; Mukherjee, D. J. Chem. Phys. 1999, 110, 6171–6188. (99) Hanrath, M. J. Chem. Phys. 2005, 123, 084102. (100) Malrieu, J. P.; Daudey, J. P.; Caballol, R. J. Chem. Phys. 1994, 101, 8908–8921. (101) Meller, J.; Malrieu, J. P.; Caballol, R. J. Chem. Phys. 1996, 104, 4068–4076. ˘ (102) Huba˘c, I.; Pittner, J.; Cársky, P. J. Chem. Phys. 2000, 112, 8779–8784. (103) Maitra, R.; Sinha, D.; Mukherjee, D. J. Chem. Phys. 2012, 137, 024105. (104) Mukherjee, D. Int. J. Quantum Chem. 1986, 30 (S20), 409–435. (105) Mukhopadhyay, D.; Datta, B.; Mukherjee, D. Chem. Phys. Lett. 1992, 197, 236–242. (106) Koch, S. Theor. Chim. Acta 1991, 81, 169–183. (107) Meissner, L. Chem. Phys. Lett. 1996, 255, 244–250. (108) Landau, A.; Eliav, E.; Kaldor, U. Chem. Phys. Lett. 1999, 313, 399–403. (109) Landau, A.; Eliav, E.; Ishikawa, Y.; Kaldor, U. J. Chem. Phys. 2004, 121, 6634–6639. (110) Malrieu, J. P. Mol. Phys. 2013, 111, 2451–2462. 27 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(111) Zaitsevskii, A.; Malrieu, J. P. Chem. Phys. Lett. 1995, 233, 597–604. (112) Zaitsevskii, A.; Malrieu, J. P. Chem. Phys. Lett. 1996, 250, 366–372. (113) Heully, J. L.; Malrieu, J. P.; Zaitsevskii, A. J. Chem. Phys. 1996, 105, 6887–6891. (114) MahaPatra, U. S.; Datta, B.; Mukherjee, D. J. Phys. Chem. A 1999, 103, 1822–1830. (115) MahaPatra, U. S.; Datta, B.; Mukherjee, D. Chem. Phys. Lett. 1999, 299, 42–50. (116) P.Ghosh,; Chattopadhyay, S.; Jana, D.; Mukherjee, D. Int. J. Mol. Sci. 2002, 3, 733–754. (117) Mao, S.; L. Cheng, W. L.; Mukherjee, D. J. Chem. Phys. 2012, 136, 024105. (118) Mao, S.; Cheng, L.; Liu, W.; Mukherjee, D. J. Chem. Phys. 2012, 136, 024106. (119) Haunschild, R.; Mao, S.; Mukherjee, D.; Klopper, W. Chem. Phys. Lett. 2012, 531, 247– 251. (120) Haunschild, R.; Cheng, L.; Mukherjee, D.; Klopper, W. J. Chem. Phys. 2013, 138, 211101. (121) Sen, A.; S. Sen, P. K. S.; Mukherjee, D. J. Comput. Chem. 2015, 36, 670–688. (122) Sen, A.; Sen, S.; Mukherjee, D. J. Chem. Theory Comput. 2015, 11, 4129–4145. (123) Chattopadhyay, S.; Pahari, D.; Mukherjee, D.; Mahapatra, U. S. J. Chem. Phys. 2004, 120, 5968–5986. (124) Hoffmann, M. R. J. Phys. Chem. 1996, 100, 6125–6130. (125) Khait, Y. G.; Song, J.; Hoffmann, M. R. J. Chem. Phys. 2002, 117, 4133–4145. (126) Jiang, W.; Khait, Y. G.; Hoffmann, M. R. J. Phys. Chem. A 2009, 113, 4374–4380. (127) Sherrill, C. D.; Schaefer III, H. F. Adv. Quantum Chem. 1999, 34, 143–269. (128) Meyer, W. In Methods of Electronic Structure Theory; Schaefer III, H. F., Ed.; Modern Theoretical Chemistry; Plenum Press: New York, 1977; Vol. 3; pp 413–446. 28 ACS Paragon Plus Environment

Page 28 of 35

Page 29 of 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(129) Siegbahn, P. E. M. Int. J. Quantum Chem. 1980, 18, 1229–1242. (130) Werner, H. J.; Reinsch, E. A. J. Chem. Phys. 1982, 76, 3144–3156. (131) Werner, H. J. J. Chem. Phys. 1988, 89, 5803–5814. (132) Knowles, P. J.; Werner, H. J. Theor. Chim. Acta 1992, 84, 95–103. (133) Shamasundar, K. R.; Knizia, G.; Werner, H. J. J. Chem. Phys. 2011, 135, 054101. (134) Wang, Y.; Han, H.; Lei, Y.; Suo, B.; Zhu, H.; Song, Q.; Wen, Z. J. Chem. Phys. 2014, 141, 164114. (135) Saitow, M.; Kurashige, Y.; Yanai, T. J. Chem. Phys. 2013, 139, 044118. (136) Wu, W.; Song, L.; Cao, Z.; Zhang, Q.; Shaik, S. J. Phys. Chem. A 2002, 106, 2721–2726. (137) Song, L.; Wu, W.; Zhang, Q.; Shaik, S. J. Comput. Chem. 2004, 25, 472–478. (138) Liu, W.; Hoffmann, M. R. Theor. Chem. Acc. 2014, 133, 1481. (139) Nakatsuji, H.; Ehara, M. J. Chem. Phys. 2005, 122, 194108. (140) Daudey, J.-P.; Heully, J.-L.; Malrieu, J.-P. Chem. Phys. Lett. 1993, 99, 1240–1254. (141) Heully, J.-L.; Malrieu, J. P.; Nebot-Gil, I.; Sanchez-Marin, J. Chem. Phys. Lett. 1996, 256, 589–594. (142) Gdanitz, R. J.; Ahlrichs, R. Chem. Phys. Lett. 1988, 143, 413–420. (143) Szalay, P. G.; Bartlett, R. J. Chem. Phys. Lett. 1993, 214, 481–488. (144) Szalay, P. G. Chem. Phys. 2008, 349, 121–125. (145) Khait, Y. G.; Jiang, W.; Hoffmann, M. R. Chem. Phys. Lett. 2010, 493, 1–3. (146) Wang, Y.; Suo, B.; Zhai, G.; Wen, Z. Chem. Phys. Lett. 2004, 389, 315–320.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(147) Siegbahn, P. E. M. Chem. Phys. 1977, 25, 197–205. (148) Siegbahn, P. E. M. Int. J. Quantum Chem. 1983, 23, 1869–1889. (149) Giner, E.; Scemama, A.; Caffarel, M. Can. J. Chem. 2013, 91, 879–885. (150) Giner, E.; Scemama, A.; Caffarel, M. J. Chem. Phys. 2015, 142, 044115. (151) Scemama, A.; Applencourt, T.; Giner, E.; Caffarel, M. J. Chem. Phys. 2014, 141, 244110. (152) Epstein, P. S. Phys. Rev. 1926, 28, 695–710. (153) Nesbet, R. K. Proc. Roy. Soc. A 1955, 230, 312–321. (154) Davidson, E. R. J. Comput. Phys. 1975, 17, 87–94. (155) MacDonald, J. K. L. Phys. Rev. 1933, 43, 830–833. (156) Hylleraas, E. A.; Undheim, B. Z. Phys. 1930, 65, 759–772. (157) Nakatsuji, H. J. Chem. Phys. 2000, 113, 2949–2956. (158) Nakatsuji, H.; Davidson, E. R. J. Chem. Phys. 2001, 115, 2000–2006. (159) Nakatsuji, H.; Ehara, M. J. Chem. Phys. 2002, 117, 9–12. (160) Dunning Jr., T. H. J. Chem. Phys. 1989, 90, 1007–1023. (161) Olsen, J.; Jørgensen, P.; Koch, H.; Balkova, A.; Bartlett, R. J. J. Chem. Phys. 1996, 104, 8007–8015. (162) Khait, Y. G.; Jiang, W.; Hoffmann, M. R. Chem. Phys. Lett. 2010, 493, 1–10. (163) Huzinaga, S.; Arnau, C. J. Chem. Phys. 1971, 54, 1948–1951. (164) Davidson, E. R. J. Chem. Phys. 1972, 57, 1999–2005.


Page 30 of 35

Page 31 of 35

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(165) Potts, D. M.; Taylor, C. M.; Chaudhuri, R. K.; Freed, K. F. J. Chem. Phys. 2001, 114, 2592–2600. (166) Barr, T. L.; DaVidson, E. R. Phys. Rev. A 1970, 1, 644–658. (167) Meyer, W. Int. J. Quantum Chem. 1971, 5 (S5), 341–348. (168) Meyer, W. J. Chem. Phys. 1973, 58, 1017–1035. (169) Ahlrichs, R.; Driessler, F. Theoret. Chim. Acta 1975, 36, 275–287. (170) Neese, F.; Wennmohs, F.; Hansen, A. J. Chem. Phys. 2009, 130, 114108. (171) Ivanic, J.; Ruedenberg, K. Theor. Chem. Acc. 2001, 106, 339–351. (172) Rolik, Z.; Szabados, A.; Surján, P. R. J. Chem. Phys. 2008, 128, 144101.



Figure 1: Deviations of iCI (with secondary states) from FCI for cc-pVDZ H2 O.

-1

1 .0 2 .0 1 .0 2 .0 2 .5 3 .0

-2

lo g a r ith m ic e r r o r in E h

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

-3

R e , R e , R e , R e , R e , R e ,

H F H F C A S C A S C A S C A S

(4 ,4 (4 ,4 (4 ,4 (4 ,4

) ) ) )

-4

-5

-6 0

1

2

3

4

5

6

7

8

ite r a tio n


9

1 0

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Figure 2: Deviations of iCI (without secondary states) from FCI for cc-pVDZ H2 O.

-1

1 .0 2 .0 1 .0 2 .0 2 .5 3 .0

-2


Page 33 of 35

-3

R e , R e , R e , R e , R e , R e ,


(4 ,4 (4 ,4 (4 ,4 (4 ,4

) ) ) )

-4

-5

-6 0

1

2

3

4

5

6

7

8

9

1 0

ite r a tio n


1 1

1 2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 1: Deviations (in a.u.) of iCI, MRCISD, MRCISD(TQ) and ec-CCSDt-CASSCF from FCI/DMRG for equilibrium water. The numbers of macro- and micro-iterations of iCI are in parentheses. iCI

cc-pVDZ

cc-pVTZ

(0,0)a

0.165833

0.237142

(1,0)

0.006786

0.022345

(1,1)

0.004894

0.011699

(1,2)

0.010055

(2,0)

0.000219

0.000821

(2,1)

0.000061

0.000089

(2,2) Extrapolated

-0.000056 -0.000088b -0.000527c

MRCISD

0.004681

0.009461

MRCISD(TQ)

0.000084

0.000363

0.00032

0.00088

ec-CCSDt-CASSCFd FCI

-76.241860e

a

SA-CASSCF.

b

Based on (0,0), (1,1) and (2,1).

c

Based on (0,0), (1,2) and (2,2).

d

Ref. 67

e

Ref. 161

f

DMRG(2000) result. 67

-76.3440f


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For Table of Contents Only

-1

1 .0 2 .0 1 .0 2 .0 2 .5 3 .0

-2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-3

R e , R e , R e , R e , R e , R e ,


(4 ,4 (4 ,4 (4 ,4 (4 ,4

) ) ) )

-4

-5

-6 0

1

2

3

4

5

6

7

ite r a tio n


8

9

1 0

iCI: Iterative CI toward full CI - American Chemical Society

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