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Alkali and Alkaline Earth Metal Compounds Core-Valence Basis Sets and Importance of Subvale

Alkali and Alkaline Earth Metal Compounds Core-Valence Basis Sets and Importance of Subvale
Alkali and Alkaline Earth Metal Compounds Core-Valence Basis Sets and Importance of Subvale

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Alkali and Alkaline Earth Metal Compounds:Core-Valence Basis Sets and Importance of Subvalence Correlation Mark A.Iron,Mikhal Oren,and Jan M.L.Martin Department of Organic Chemistry,Weizmann Institute of Science,76100Reh .ovot,Israel ?(Dated:Mol.Phys.,in press (W.G.Richards special issue))Abstract Core-valence basis sets for the alkali and alkaline earth metals Li,Be,Na,Mg,K,and Ca are proposed.The basis sets are validated by calculating spectroscopic constants of a variety of di-atomic molecules involving these elements.Neglect of (3s,3p )correlation in K and Ca compounds will lead to erratic results at best,and chemically nonsensical ones if chalcogens or halogens are present.The addition of low-exponent p functions to the K and Ca basis sets is essential for smooth convergence of molecular properties.Inclusion of inner-shell correlation is important for accurate spectroscopic constants and binding energies of all the compounds.In basis set extrapo-lation/convergence calculations,the explicit inclusion of alkali and alkaline earth metal subvalence correlation at all steps is essential for K and Ca,strongly recommended for Na,and optional for Li and Mg,while in Be compounds,an additive treatment in a separate ‘core correlation’step is probably su?cient.Consideration of (1s )inner-shell correlation energy in ?rst-row elements requires inclusion of (2s,2p )‘deep core’correlation energy in K and Ca for consistency.The latter requires special CCV n Z ‘deep core correlation’basis sets.For compounds involving Ca bound to electronegative elements,additional d functions in the basis set are strongly recommended.For optimal basis set convergence in such cases,we suggest the sequence CV(D+3d )Z,CV(T+2d )Z,CV(Q+d )Z,and CV5Z on calcium.

I.INTRODUCTION

The rate-determining step in accurate wavefunction ab initio calculations is the determi-nation of the correlation energy.Moderately reliable methods like CCSD(coupled cluster with all single and double substitutions[1])have theoretical CPU time scalings∝n2N4, where n and N represent the numbers of electrons and basis functions,respectively.For the CCSD(T)(i.e.,CCSD with a quasiperturbative estimate of the e?ect of connected triple substitutions[2])method—which has been termed the“gold standard of quantum chem-istry”(T.H.Dunning)—the corresponding scaling is∝n3N4.

For this reason,the idea of only including the‘chemically relevant’valence electrons in the correlation problem and constraining the inner-shell or‘core’orbitals to be doubly occupied gained currency early on.Bauschlicher et al.[3]were the?rst to propose a partitioning of the inner-shell correlation energy into a core-core(CC)component—involving excitations exclusively out of the inner-shell orbitals—and a core-valence(CV)component—involving simultaneous excitations out of the valence and inner-shell orbitals.While,at least in elements with a large core-valence gap(see below),the CC component can be expected to roughly cancel between a molecule and its separated atoms,the CV component may a?ect chemically relevant molecular properties to some degree.

Over time,it has become recognized that the inclusion of inner-shell correlation—even in?rst-row systems—is important for accurate binding energies(e.g.[4,5]),molecular geometries(e.g.[4]),and harmonic frequencies(e.g.[4]).The treatment of inner-shell cor-relation requires special basis sets which not only have additional radial?exibility,but also include‘hard’or‘tight’(i.e.,high-exponent)d,f,...functions in order to cover angular correlation from the inner-shell orbitals.Special basis sets of this type have been developed, such as the cc-pCV n Z(correlation consistent polarized core-valence n-tuple zeta)basis sets of Woon and Dunning[6]for B–Ne,the MT(Martin-Taylor[7])and MTsmall[8]basis sets for Li–Ar,the rather small G3large basis set[9]of Pople and coworkers for the main group elements of the?rst three rows,and very recently the cc-pwCV n Z(i.e.core-valence weighted cc-pCV n Z)basis sets of Peterson and Dunning[10]for B–Ne and Al–Ar.

While the CC+CV contribution to absolute correlation energies may rival or exceed the valence contribution,its di?erential contribution to the molecular binding energy is gener-ally small compared to the SCF and valence correlation contributions(typically less than

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1%for1st-row systems).For that reason—as well as the formidable cost of core corre-lation calculations—computational thermochemistry schemes(e.g.G3theory[9],W1/W2 theory[8,11])that account for inner-shell correlation generally treat the latter as a small additive contribution obtained with a relatively compact core correlation basis set.(The ?nding[8]that connected triple substitutions are surprisingly important for CV contribu-tions ensures that the core correlation step often is the rate-determining one for benchmark thermochemistry calculations,particularly beyond the?rst row of the Periodic Table.)Re-cently,some attempts have been made to replace the core correlation step by bond additivity approximations[12,and references therein],and promising results have been obtained[13]by means of core polarization potentials[14].

Despite repeated warnings in the literature against the practice(e.g.,by Taylor[15],by one of the present authors[4],and by Woon and Dunning[6]),core correlation is often included —for technical reasons or‘because it cannot hurt’—in calculations using basis sets(e.g. the standard Dunning correlation consistent basis sets[16,17],which are of minimal basis set quality in the inner-shell orbitals)which are not adapted for nonvalence correlation.This can often cause errors on computed properties well in excess of the basis set incompleteness error:For instance,a comparison between valence-only[18]and all-electron[19]CCSD(T)/cc-pVTZ harmonic frequency calculations on the cyclic isomer of C4suggests core correlation contributions to the harmonic frequencies of up to50cm?1,a factor of?ve more than the true correction obtained[18]with a core correlation basis set.

For the elements B–Ne and Al–Ar,the gap between valence and inner-shell orbital energies is large enough that a conventional core-valence separation of the orbital energies is easily made.Things are less simple further down the periodic table.Bauschlicher[20,21]noted that for gallium and indium halides,the valence(2s)orbitals of the halogen are below the Ga(3d)or In(4d)‘core’orbitals in energy.At the very least,these d orbitals should be added to the correlation space;suitable basis sets for this purpose have been developed by Martin and Sundermann[22]for Ga and Ge,and by Bauschlicher[21]for In.

But similar issues arise with Group1and2elements.Radom and coworkers[23,24] recently noted catastrophic failures of G2theory and related methods in predicting the heats of formation of various alkali and alkaline earth metal oxides and hydroxides:for example,the binding energy of K2O is underestimated by no less than256kJ/mol!Upon correlating all electrons,this error decreases to60kJ/mol,which is further decreased to17

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kJ/mol when more sophisticated electron correlation methods are used.Inspection of an atomic orbital energy table[25]quickly reveals why:the oxygen2s and2p valence orbitals lie below the potassium3s and3p inner-shell orbitals,rendering the conventional separation between‘valence’and‘core’orbitals essentially meaningless.Inclusion of Group1and2 subvalence correlation—termed‘relaxed inner valence’(RIV)by Radom and coworkers[23]—may in fact be the appropriate treatment,not just for K(and Ca)compounds,but for alkali and alkaline earth metal compounds in general.(One of the?rst papers to recognize the importance of inner-shell correlation for alkali metals may have been that by Liu and coworkers.[26])

This problem is far from academic,considering the great importance of K+,Ca2+,Na+, and Mg2+complexes in molecular biology.In addition,a number of these interactions(e.g., of the cation-πtype)are not necessarily amenable to density functional treatments without validation of some kind by means of ab initio methods.Clearly,the availability of high-quality core-valence basis sets for the elements Li,Be,Na,Mg,K,and Ca would bene?t the high-accuracy computational thermochemistry and spectroscopy communities as well as biomolecular modelers.The purpose of the present work is the development and validation of CVnZ(core-valence n-tuple zeta,n=D,T,Q,5)basis sets for these elements.

https://www.doczj.com/doc/d28720733.html,PUTATIONAL DETAILS

All electronic structure calculations carried out in the present work were carried out using the MOLPRO2000.1and2002.3program systems[27,28]running on SGI Origin2000(12×MIPS R10000,300MHz and4×MIPS R10000,195MHz,IRIX6.5)and Compaq ES40 (4×EV67,667MHz,Tru64Unix4.0f)minisupercomputers,as well as Compaq XP1000 (EV6,500MHz,Tru64Unix4.0f)and dual Intel Xeon(1.7,2.0and2.4GHz,RedHat Linux 7.2and7.3)workstations.

Unless explicitly noted otherwise,energy calculations were carried out at the CCSD(T) level;for open-shell systems,single-determinant ROHF reference functions were used,but the de?nition of the CCSD(T)energy according to Watts et al.[29]was employed throughout.

Validation calculations on diatomic molecules were carried out as follows.Energies were computed at21equidistant points around the putative bond distance with interpoint spac-ings of0.01–0.03?A,depending on the curvature of the surface.Energies at these points

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were converged as precisely as feasible,with integral evaluation thresholds being tightened as necessary.Polynomials of increasing order were then?t through the points,and the signi?cance of the additional order terms subjected to a Fisher-Snedecor test.The expan-sion was truncated at the point where the two-way signi?cance of the next higher order dropped below99%.The retained expansion was generally of order between six and eight. The minimum of this curve was then sought by means of a Newton-Raphson method,and the curve re-expanded around the minimum.A Dunham analysis[30]was then carried out on the?nal curve.The computed dissociation energies at0K,D0,reported in the tables include anharmonic zero-point corrections from the computedωe andωe x e at the same level of theory.

As for the accompanying basis sets,standard cc-pV n Z basis sets were used throughout on H.For O and F,aug-cc-pV n Z basis sets[31]were used in valence calculations and aug-cc-pCV n Z basis sets[6]in calculation where the O and F(1s)cores were correlated.For S and Cl,the aug-cc-pV(n+d)Z basis sets of Wilson et al.[32]were used in valence calculations, and the cc-pCV n Z basis sets of Peterson[10]in calculations where the(2s,2p)cores of these elements were correlated.

Where deemed necessary,scalar relativistic e?ects were assessed by means of CCSD(T) calculations within the Douglas-Kroll-Hess approximation[33,34]as implemented in MOL-PRO2002.3[28].

III.OPTIMIZATION OF CV n Z BASIS SETS

A.General procedure

The basis set optimizations were carried out using a Fortran program developed in-house. Derivatives were obtained numerically using central di?erences,yielding both the gradient and the diagonal of the Hessian.The step thus obtained—essentially Newton-Raphson with neglect of o?-diagonal Hessian elements—was combined with a Brent line search.Our experience with this simple but fairly robust optimization algorithm is quite good as long as optimization parameters are not too strongly coupled.In the presence of signi?cant coupling, DIIS(Direct Inversion of the Iterative Subspace)[35]dramatically sped up convergence.

Finite di?erence step sizes were made roughly proportional to the parameters themselves,

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and increased as necessary if the parameter surface in the a?ected direction was found to be particularly‘?at’.

Some additional optimizations were carried out using an adaptation of the DOMIN program by P.Spellucci[36],which is an implementation of the BFGS(Broyden-Fletcher-Goldfarb-Shanno)variable-metric method.Numerical derivatives of order two,four,and six were used:the lower orders until an approximate minimum was reached,after which the optimization was re?ned using the higher orders.In most cases,however,the simple procedure outlined above appeared to be more robust.

For the purposes of the optimization(and since only very small systems are involved), integral evaluation,SCF and CI convergence thresholds in MOLPRO were tightened to essentially machine precision.Basis set parameters were converged to at least four signi?cant digits,and?ve where at all possible.

In cases where many primitives of a particular angular symmetry are required,we initially constrained these functions to follow an even-tempered sequenceζi=αβ(i?n?1)/2,where n is the total number of primitives.Only when optimumαandβhad been reached were the individual exponents optimized further without any constraints.If a set of(n?1)primitives of that symmetry was already available(in casu,from a previously optimized smaller basis set),the geometric mean of their exponents was taken as the starting value forα,and the arithmetic mean of the ratios between successive primitives as the initial value forβ.

A few‘checks and balances’can generally be applied to the?nal exponents:

1.within a row,the geometric mean of the exponents should increase roughly as(Z?)2,

where Z?is the‘shielded’nuclear charge(?rst Ahlrichs-Taylor rule[37]);

2.exponents in a larger set(e.g.3d)should mesh with the next smaller set(e.g.2d);

3.the ratio of successive exponents in a set should be about two or higher,and all gaps

within the core or valence parts should roughly be of the same order;

4.the geometric mean of the exponents for each angular momentum should roughly

increase by a factor of1.2for each step up in L(second Ahlrichs-Taylor rule[37]).

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B.Li and Be

For Li and Be,the‘uno?cial’cc-pV n Z basis sets of Feller[38]supplied in the MOLPRO 2000.1basis set library were taken as starting points.The quantity optimized for is the(1s) correlation energy at the CISD level for the Li(2S)and Be(1S)ground states,respectively. This happens to be equivalent to E[CISD,full]?E[SCF]for Li,but is equal to E[CISD,full]-E[CISD,valence]for Be.

There exists a tradition of optimizing basis sets for correlated calculations at the CISD level,since CISD is a variational method.However,optimizations for the CCSD or CCSD(T) counterparts of the abovementioned correlation energies yield very similar basis sets,which would have been of identical quality in molecular calculations.(A table of these exponents for the CVDZ,CVTZ,and CVQZ cases can be found in the Supplementary Material[39].) By analogy with the cc-pCV n Z basis sets for B–Ne[40],1s1p,2s2p1d,3s3p2d1f,and 4s4p3d2f1g sets of primitives were added to the cc-pVDZ,cc-pVTZ,cc-pVQZ,and cc-pV5Z basis sets,respectively.Successive angular momenta were optimized individually,after which the complete set of exponents was further optimized together.

Multiple local minima exist for the larger sets,and care was taken to ensure that the ?nal optimized basis set re?ects the most‘contiguous’(or most‘even-tempered’)solution in the sense of having no obvious‘gaps’between exponents.

Aside from this issue,the basis set optimizations proceeded uneventfully.

C.Na and Mg

Likewise,for Na and Mg,unpublished cc-pV n Z basis sets of Feller[38]were taken as the starting point.In this case,we optimized for the2s2p correlation energy,found as E[CISD,2s2p3s]-E[CISD,3s]for Mg and as simply E[CISD,2s2p3s]-E[SCF]for Na.Note that the1s orbitals are constrained to be doubly occupied throughout:not only will the e?ect of this’deep core’orbital on chemically signi?cant properties be negligible,but including it in the optimization would bias the exponents towards the large—but chemically quite invariant—core-core correlation energy rather than the chemically more signi?cant2s2p core-core and core-valence correlation energies.

Since the highest core correlation orbital is of p symmetry in this case,the basis sets

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are rather larger,involving addition of1s1p1d,2s2p2d1f,3s3p3d2f1g,4s4p4d3f2g1h core correlation sets to the cc-pVDZ,cc-pVTZ,cc-pVQZ,and cc-pV5Z sets,respectively.

For the CV5Z basis sets for Na and Mg,simply optimizing a set of core correlation s functions leads to such serious near-linear dependence problems that the basis would in practice be unusable.For want of an alternative,we simply uncontracted an additional four s primitives instead.

D.K and Ca

The core-valence gap in Ca,and particularly K,is so small that the optimization of regular cc-pV n Z basis sets would be of largely academic interest.Feller and coworkers previously published CVDZ,CVTZ,and CVQZ basis sets for K[41];the sp exponents for the CVDZ and CVTZ basis sets were taken from(15s,12p)and(18s,15p)basis sets,respectively,of Ahlrichs and coworkers[42],while those for the CVQZ basis set were taken from the Partridge-1 set[43],which is of(23s,19p)uncontracted size.

Initially we employed these basis sets unaltered,and merely added a CV5Z basis set. This was obtained as follows,starting from the Partridge-3basis set:

?the basis set was contracted to miminal(i.e.[4s3p])in an atomic calculation on the ground state

?the four outermost s and p primitives each were decontracted

?[4d3f2g1h]functions were optimized at the valence-only CISD level for the K2molecule at its experimental bond distance,yielding an intermediate cc-pV5Z basis set

?core correlation functions were obtained by optimizing the atomic E[CISD,3s3p4s]-E[SCF]energy.The optimization of the s and p functions was plagued by insur-mountable near-linear dependence problems,and we ended up merely decontracting an additional four primitives of s and p symmetries each.In addition,a crossover occurred between the highest d exponent of the underlying cc-pV5Z basis set and the lowest d exponent required for the inner-shell correlation,and it was decided to merely keep the outermost three d primitives constant while optimizing all?ve remaining d primitives for3s3p correlation.

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Thus,our original CV5Z basis set was obtained.However,as evidenced in Table I, basis set convergence of properties for K2and KH was less than satisfactory.In particular, inspection of the dissociation energy as a function of the basis set reveals that for K2,the CVTZ,CVQZ,and CV5Z results are nearly collinear,which is obviously undesirable.For this reason,the d,f,...exponents of the K CVTZ and CVQZ basis sets were reoptimized in the same manner as the CV5Z basis set.In the process we found the‘valence correlation’d and f exponents in the original Feller basis sets to be excessively‘tight’,and with the adjusted basis sets(denoted‘version0.1’in Table I),increases of6.5and7.0kJ/mol are seen in the CVTZ and CVQZ D0values.Other molecular constants are also a?ected quite signi?cantly.While the Feller CVTZ and CVQZ were not found to be as problematic for the other K compounds considered in this work,they are clearly unsuitable for K2or any other system with similar long-distance bonding.

Analogously,we optimized CV n Z(n=D,T,Q,5)basis sets for Ca,starting from Ahlrichs 14s9p,Partridge-120s12p,Partridge-223s16p,and Partridge-326s16p primitive sets.Ini-tial attempts to optimize basis sets for the3P atomic excited state(analogously to the sp-hybridized state of Be)yielded exceedingly poor basis set convergence of molecular prop-erties,and these basis sets were abandoned in favor of1S ground-state optimized basis sets. (We also attempted optimizations in the CaH2molecule,and found basis set parameters obtained there to be very similar to those for the1S atomic ground state.)Valence correla-tion functions were optimized for the valence correlation energy of the ground-state calcium atom,and core-valence functions for E[CISD,3s3p4s]-E[CISD,4s].For the CVQZ and CV5Z basis sets,the optimized‘valence’and‘subvalence’d and f shells interlock,and as a result only the outermost2d2f were held at their valence-optimized values and the remainder reoptimized for inner-shell correlation.No such issues arose with the g and h functions. Once again,basis set convergence of molecular properties(e.g.for CaH,but also for other diatomics,not displayed here)was found to be somewhat unsatisfactory.

For both atoms,we then optimized a sequence of four‘stretch-tuned’basis sets[44], i.e.basis sets in which the exponents of each angular momentum obey the following four-parameter relation:

lnζn=α+n(β+(n?1)(γ+(n?2)δ))(1) (A total of only eight parameters thus had to be optimized for each basis set.)The particular sequence of contraction sizes chosen was(20s12p),(22s14p),(24s16p),and(26s18p).The

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valence and inner-shell correlation functions were optimized as before.

These basis sets(denoted‘version0.2’in Table I)appeared to have better convergence properties.However,something was clearly still not satisfactory.For instance,basis set convergence for the ionization potentials of K and Ca atoms was found to be atypically slow (Table II).

It then occurred to us that,as the4s orbital of these atoms is very di?use,the atom-optimized p functions may be too‘tight’(as they were optimized for atoms with vacant 4p orbitals)to contain suitable?rst polarization/angular correlation functions for the outer part of the4s orbital.This assumption was veri?ed by optimizing single‘probe’basis functions of various symmetries on top of CV n Z basis sets held constant.(Only valence electrons were correlated,as otherwise the functions would have fallen into the‘gravity well’of the inner-shell electrons.)A very noticeable lowering of the total energy was seen upon adding a single p function,the optimum exponent of which indeed turned out to be ‘looser’than any of the existing p functions.An additional‘loose’p function led to a further improvement for the K CV n Z(n=D,T,Q)basis sets,but not for the K CV5Z basis set(the outer p primitive of which is already quite‘loose’)or any of the Ca basis sets.Consequently, the valence correlation parts of the CV n Z basis sets were reoptimized with two and one additional p functions added,respectively,for K and Ca,and afterwards the inner-shell part was reoptimized for consistency.Thus our?nal CV n Z basis sets were obtained.As seen in Table I,basis set convergence of molecular parameters is now satisfactory.For IP(K)and IP(Ca),nearly exact values are now obtained(Table II).

Since the primitive K and Ca basis sets may be somewhat bulky for some applications, we have also generated SDB-CV n Z basis sets,which are combinations of the valence and subvalence parts of the CV n Z basis sets with small-core Stuttgart-Dresden-Bonn[45]rel-ativistic e?ective core potentials.The latter replace the(1s2s2p)electrons.Unlike the situation with large-core basis set where reoptimization of the basis set is essential[22],we have kept all exponents at their all-electron values and merely‘pruned’and recontracted the basis set.Speci?cally,the atomic SCF calculation was repeated with the(1s2s2p)orbitals replaced by the SDB pseudopotential denoted‘ECP10MWB’.Then all primitives with ab-solute coe?cients of less than10?5in any valence orbitals were deleted,and the orbitals recontracted.

All basis sets obtained in this work are available in machine-readable form(Gaussian and

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MOLPRO formats)as supplementary material[39]to the present paper.Contracted basis set sizes and numbers of basis functions for each element are given in Table IV.

IV.V ALIDATION FOR DIATOMIC MOLECULES

A.Diatomic metal hydrides

Since hydrogen has no core electrons,there are no di?erential core-core contributions (just core-valence)to the molecular https://www.doczj.com/doc/d28720733.html,puted and experimental data for the diatomic hydrides are given in Table V.

As noted previously,the inner-shell contribution to the LiH spectroscopic constants is quite appreciable,as can be expected from the small Li(1s)-H(2s)gap of about2a.u.We note that the CCSD(T,riv)/CV5Z results reproduce the Born-Oppenheimer bond distance and harmonic frequency to within0.0002?A and0.2cm?1,respectively.Neglect of Li(1s)cor-relation leads to errors of0.012?A and13.5cm?1,respectively.Nevertheless,the contribution to D e does not exceed1.2kJ/mol.

The corresponding core-valence gap for BeH is almost doubled,which translates into sub-stantially reduced core-valence contributions(relatively speaking)to r e andωe.Interestingly, the contribution to D0reaches nearly2kJ/mol.

In contrast,the Na(2p)-H(1s)gap narrows to no more than1a.u.,and in NaH we see inner-shell correlation contributions of33cm?1toωe and0.035?A to r e.Interestingly,once more D e is nearly una?ected.Once more,the CCSD(T,riv)/CV5Z results are in excellent agreement with experiment for the spectroscopic constants:applying a W2-type extrapola-tion for D0results in a value of182.35kJ/mol,within0.3kJ/mol of experiment.

Given that the Mg(2p)-H(1s)gap is not dissimilar from the corresponding one in LiH, it is not greatly surprising that the importance of core-valence contributions is again much smaller than in NaH.However,the8kJ/mol contribution to D0is considerably more sig-ni?cant.

In KH,the K(3p)and H(1s)orbitals are closer than0.5a.u.,and the contributions of nearly0.1?A and44cm?1to r e andωe,respectively,clearly suggest that any sort of‘valence correlation only’calculation on a K compound should be viewed with great suspicion.Note that the core-valence contribution to D0is negative in this case,as was previously found

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(e.g.[12])for aluminum and silicon hydrides.

In comparison,the Ca(3p)–H(1s)gap widens to0.84a.u.,and core-valence e?ects on the molecular properties are mitigated accordingly.Like for MgH and KH,inner-shell correlation in fact slightly reduces the dissociation energy.

With the exception of KH and CaH,the di?erential core-valence contributions appear to have converged with respect to the basis set at the CVQZ stage.This in itself satis-?es a minimum requirement for treating the inner-shell correlation contribution separately, something which is de?nitely inappropriate for KH.

On the whole,agreement between CCSD(T,riv)/CV5Z and experiment can only be de-scribed as quite satisfactory.For systems other than LiH and NaH,imperfections in the CCSD(T)electron correlation method may account for most of the remaining discrepancy between computed and observed harmonic frequencies[46].

B.Metal diatomics

The alkaline earth metal dimers(particularly Be2)exhibit such severe multireference character that they warrant studies in themselves(which would,however,focus on electron correlation methods rather than basis sets,see e.g.[47]and references therein).Results for species other than Be2can be found in Table VI.

In Li2,subvalence correlation accounts for0.024?A,which is sizable by spectroscopist’s standards.In Na2,this is drastically increased to nearly0.1?A,and reaches a whopping 0.23?A in K2.Interestingly,these contributions are quite close to double their metal hydride counterparts.As expected,LiNa and NaK represent scenarios intermediate between the homonuclear diatomics of the constituent elements.Changes inωe are modest in absolute numbers,but for these molecules represent relative errors of up to5%.It is quite clear how-ever that,with the possible exception of Li2,inclusion of subvalence correlation is essential for reliable molecular parameters.Once again we see,however,that bond energies are only a?ected mildly:0.8kJ/mol in Li2,1.7kJ/mol in LiNa,1.6kJ/mol in Na2,and only0.5 kJ/mol in K2.

Once again,a basis set of CVQZ quality appears to be close to convergence for the di?erential core-valence e?ects.

E?ects for Mg2and Ca2are considerably milder than for their alkali neighbors,yet(3s,3p)

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correlation reduces the Ca2bond distance by0.07–0.08?A,not negligible by any reasonable standard.The discrepancies between theory and experiment for Mg2primarily re?ect the inadequacy of the CCSD(T)method for this highly multireference system.In contrast,Ca2 yields quite satisfying results compared to experiment.

Agreement between our best calculations and experiment can only be described as excel-lent for Li2.The same would be true of Na2if it were not for the bond length which is still 0.003?A too long at the CCSD(T,riv)/CV5Z level.However,at the DK-CCSD(T)/CVQZ level,we?nd a scalar relativistic correction to the bond length of-0.006?A.(If this correc-tion would seem to be exaggerated,we note that the Na2potential curve is quite?at,and that already for OH?and HF,scalar relativistic corrections were found to be required for spectroscopic-quality results[48].)In K2,our best calculated results are likewise in excellent agreement with the experimental spectroscopic data except for r e,which is calculated to be 0.014?A longer than the observed value.At the DK-CCSD(T)/CVQZ level,we?nd a rela-tivistic contribution of-0.013?A to the bond length,which explains most of the discrepancy. In addition,the(2s,2p)‘deep core’orbitals are energetically in the same range as the(1s) orbitals in C and N,so it cannot be entirely ruled out that‘deep core’correlation may a?ect molecular properties in K compounds.

We optimized a CCVTZ(deep core valence triple zeta)basis set for K.In these calcu-lations,the CVTZ basis set was held constant and exponents of2s2p2d1f basis functions optimized for E[CISD,2s2p3s3p4s]?E[CISD,3s3p4s],i.e.for deep-core correlation energy taken in isolation.As can be seen in Table III,these exponents are obviously much‘tighter’, by almost an order of magnitude,than those required for subvalence correlation.Hence,any ‘deep core’result in a mere valence and subvalence CV n Z(let alone a valence-only cc-pV n Z set)should be regarded with skepticism at best.

The only change of note we see in the molecular properties when using this CCVTZ basis set for K2is a shortening of the K–K bond by another0.0018?A.We also attempted to optimize a CCVQZ basis set but ran into insurmountable near-linear-dependence prob-lems;however,the CCVTZ result should at least give an indication of the(fairly modest) magnitude of this e?ect.

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C.Diatomic metal halides

This set of molecules is particularly revelant because of the deep-lying valence s orbitals on the halogen atoms.Relevant results are collected in Table VII.

Firstly,for LiF the contribution of Li(1s)correlation to the binding energy is somewhat more noticeable(4kJ/mol).Its inclusion also causes a contraction of the bond by0.015?A and an increase inωe of12cm?1.The further e?ect of permitting F(1s)correlation is nearly negligible in comparison.The Cl(2s,2p)correlation e?ect is somewhat more noticeable but still Li(1s)correlation accounts for the lion’s share of the changes.

The F(2s)and Na(2p)orbitals are nearly degenerate,and hence one would expect valence-only calculations on NaF to be problematic,to say the least.Nevertheless,while the e?ect of Na(2s,2p)correlation on the molecular constants is quite noticeable and its inclusion clearly essential for accurate calculations,the results obtained are not outright nonsensical.Of course,considering the di?erence in electronegativity between the elements,the combination of Na+and F?would yield a more appropriate(if perhaps extreme)‘atoms in molecules’picture—and the corresponding orbital energy gap in this admittedly extreme scenario is

0.72a.u.with the Partridge3basis set.The corresponding‘ionic’gap for NaCl amounts to

1.07a.u.,explaining why also for NaCl,valence-only results are not totally unreasonable.

The person carrying out calculations on KF and KCl has no such luck.The inclusion of K(3s3p)correlation shiftsωe in KF upwards by no less than20%,and the contribution to D e is seen to be on the order of170kJ/mol.In fact,the‘valence’results are better described as referring to correlating the z component of K(3p)and freezing the F(2s),which obviously makes no chemical sense at all.The valence-computed dissociation energy of KCl agrees deceptively well with experiment,but an error of0.3?A in the bond distance and of a factor of three in the anharmonicity constant should discourage any quantum chemist from performing valence-only calculations on this type of species.

While obviously signi?cant for spectroscopic purposes,the e?ects of metal subvalence correlation in MgF and MgCl are clearly less prominent than in NaF and NaCl,respectively, and with the beryllium halides,one clearly could‘get away’with a di?erential treatment. In contrast,in CaF a calcium subvalence contribution to D0of180kJ/mol is found,not to mention+46cm?1onωe and a factor of two onωe x e.Oddly,the e?ect on the bond length is comparatively small(0.04?A).The opposite scenario is seen for CaCl,with a?r e of0.1?A

14

but otherwise not outlandishly erroneous values for the other spectroscopic constants.CaF and CaCl results resemble more the potassium halides.

And once again,the CCSD(T,riv)/CV5Z results agree as well with experiment as can reasonably be expected.

We have also considered calculations in which subvalence correlation on the halide was permitted.However,for K and Ca compounds with{C,N,O,F},the(1s)orbital of these latter elements is in fact below the K(2s,2p)‘deep core’orbitals in energy,and therefore any such calculation will require the use of CCV n Z‘deep core’basis sets on the metal.Such calculations are su?ciently costly that one might want to limit them to single-point energy calculations in high-accuracy thermodynamic work.For MCl,the most noticeable e?ect of including(2s,2p)subvalence correlation on chlorine is a contraction of r e by0.003?A.

D.The metal chalcogenides

We will primarily focus on the oxides.Results can be found in Table VIII.

Clearly,changes of13cm?1and6kJ/mol,respectively,forωe and D e of BeO suggest the importance of metal(1s)correlation in this system;contributions from O(1s)correla-tion are an order of magnitude less important.Note that the experimental data can be reproduced almost exactly at the CCSD(T,all)/CV5Z level,despite BeO having pronounced multireference character and in fact being on the borderline of applicability of CCSD(T)(see e.g.[49]).In the isovalent BeS system,Be(1s)correlation is still noticeably more important than S(2s,2p)correlation,although the di?erence is not as pronounced as its counterpart in BeO.

Changes between valence-only and RIV treatment of0.017?A in r e(MgO)and23cm?1 inωe(MgO)speak for themselves.The MgO molecule has very pronounced multireference character,and accurate reproduction of the experimental spectroscopic constants would required an elaborate multireference treatment.Subvalence correlation e?ects in MgS are rather milder.

In contrast,inspection of the computed spectroscopic constants for CaO immediately reveals that valence-only calculations on this system are essentially exercises in wasting computer time.Valence-only bond distances and dissociation energies are o?by a quarter of an?A,and a factor of six in D0.Respectable agreement with experiment can however

15

be reached by correlating the Ca(3s,3p)electrons.Obviously,the large uncertainties in the measured dissociation energies preclude really?ne comparisons with experiment:on the basis of the preceding,it can perhaps be stated that the calculated binding energies are more reliable than the experimental ones for the alkali(ne earth)metal chalcogenides.

Once again,correlating the(1s)orbital on O would require including(2s,2p)deep-core correlation in Ca,otherwise meaningless results are obtained.

E.Additional d functions on Ca

Wesolowski et al.[50],in a study of calcium oxide,observed especially poor performance for a valence-only atomic natural orbital(ANO)basis set[51].This was considerably im-proved when high-exponent d functions were added.They noted that Ca+has a low-lying 2D state only1.7eV above the2S ground state,and showed that said ANO basis set over-estimates this separation by nearly a factor of three.Addition of several very high-exponent d functions,taken from the6-311G(2df)basis set for Ca[52](which treats the(3d)orbital on a valence footing),results in an at least semiquantitatively correct result.

As the present basis sets were developed from the ground up with inner-shell correlation in mind,they intrinsically contain high-exponent d functions and should therefore be much less susceptible to the problem.As shown in Table IX,quite good agreement with the experimental transition energy is obtained for CVQZ and CV5Z basis sets,but the smaller basis sets leave something to be desired.

We therefore proceeded to optimize CV(n+d)Z,CV(n+2d)Z,and CV(n+3d)Z basis sets in the following fashion.(Both the nomenclature and the procedure bear resemblance to the cc-pV(n+d)Z basis sets for second-row elements[32].)All angular momenta other than d were kept constant,as were the valence-optimized d functions(see above).The remaining d functions were reoptimized for inner-shell correlation with one,two,or three additional d functions added.(In the CV(Q+d)Z and CV(Q+2d)Z basis sets,it was found necessary to constrain the exponents to stretch-tuned and even-tempered sequences,respectively.) As can be seen in Table IX,the2D←2S excitation energy indeed converges considerably faster for the CV(n+d)Z and CV(n+2d)Z series than for the CV n Z series.After applying a scalar relativistic correction by means of the Douglas-Kroll method in an uncontracted CV(Q+d)Z basis sets,very good agreement with experiment can be achieved.As for the

16

molecular properties of CaO(Table VIII),the e?ect of additional d functions is quite dra-matic in the CVDZ case and still quite important in the CVTZ case,but tapers o?for CVQZ and becomes quite insigni?cant for CV5Z.We noticed similar behavior in the other CaX systems considered in this paper(Tables V,VI,and VII).Convergence as a function of the number of additional d functions appears to be approached for CV(D+3d)Z,CV(T+2d)Z, CV(Q+d)Z,and CV5Z(note particularly Table VIII).Basis set convergence of molecular properties along this sequence is clearly much smoother than for the unmodi?ed CV n Z basis sets.This is particularly true for CaO and,to a somewhat lesser extent,for CaF,least so for CaH and Ca2where basis set convergence was adequate to begin with.

V.CONCLUSIONS

Core-valence basis sets have been developed for the alkali and alkaline earth metals Li, Be,Na,Mg,K,and Ca.Validation calculations for a number of diatomics involving these systems reveal basis set convergence to be satisfactory.

The addition of low-exponent p functions to the K and Ca basis set is found to be essential for smooth basis set convergence of molecular properties.These functions accommodate angular correlation from the outer part of the4s orbital.

Valence-only calculations on K and Ca chalcogenides and halides yield large errors at best, and chemically nonsensical results at worst.In general,inclusion of subvalence correlation in K and Ca compounds is absolutely essential for even reliable(let alone accurate)results, while for accurate calculations,subvalence correlation in Na(and,to a lesser extent,Li and Mg)should be included,preferably as a‘baseline’treatment rather than an additive correction.The latter appears to be su?cient for Be compounds.Any study that seeks to address the importance of?rst-row element core correlation in K and Ca compounds should also consider deep-core(2s,2p)correlation from these latter elements.

For compounds involving Ca bound to highly electronegative elements,we strongly rec-ommend basis sets with additional d functions on Ca:our suggested sequence for basis set convergence studies would be CV(D+3d)Z,CV(T+2d)Z,CV(Q+d)Z,CV5Z.

17

Acknowledgments

JM is a member of the Lise Meitner Center for Computational Quantum Chemistry (Israel-Germany)and the Helen and Martin Kimmel Center for Molecular Design(Weizmann Institute).This research was supported by the Tashtiyot program of the Ministry of Science and Technology(Israel)and by the Minerva Foundation,Munich,Germany.MAI and MO acknowledge Ph.D.and M.Sc.Fellowships,respectively,from the Feinberg Graduate School (Weizmann Institute of Science).We thank a referee for bringing Ref.[50]to our attention.

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W.,Nicklass,A.,Peterson,K.,Pitzer,R.,Stone,A.J.,Taylor,P.R.,Mura,M.E.,Pulay,P., Sch¨u tz,M.,Stoll,H.,and Thorsteinsson,T.

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20

大学英语Unit 1 课文翻译

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Unit 3 近年来,随着互联网技术的迅猛发展,互联网经济已成为一个热门话题。以蓬勃发展的电子商务为代表的互联网经济已成为经济发展的重要引擎。我国政府高度重视发展互联网经济,提出了“互联网+”的概念,以推动互联网与医疗、交通、教育、金融、公共服务等领域的结合。这将为互联网经济的发展提供极大的发展潜力和更广阔的发展空间。随着“互联网+”战略的深入实施,互联网必将与更多传统行业进一步融合,助力打造“中国经济升级版”。 In recent years, with the rapid development of Internet technology, the Internet economy has become a hot issue. As represented by the promising E-commerce, the Internet economy has become a strong driving force for the economic development. Our government attaches great importance to developing the I nternet economy and proposes the concept of “Internet Plus”, aiming to integrate the Internet with other industries, such as health care, transportation, education, finance, and public service. This will create great potential and broad prospects for the development of the Internet economy. With the implementation of the “Internet Plus” strategy, the Internet is certain to be integrated with more traditional industries and help build “the upgraded version of the Chinese economy”.

地球物理勘探考点汇总

地球物理勘探知识点 一、名词解释 1.动校正:校正因炮检距不等而存在的正常时差的影响。 2.时距曲线:若测线是沿一条线进行的,则测线上各观测点坐标与波至时间的关系图称为时距曲线。 3.多次覆盖:指采用一定的观测系统获得对地下每个反射点多次重复观测的采集地震波讯号的方法。 4.电阻率剖面法:当保持供电电极距AB不动时,电极系探测深度一定,移动电极系时就可以反应一定深度范围内的地下电阻率的变化情况,这种方法称之为电阻率剖面法。 5.电法勘探:是以岩石、矿石的导电性、电化学活动性、介电性和导磁性的差异为物质基础,使用专用的仪器设备观测和研究地壳周围物理场的变化和分布规律,进而达到解决地质问题的目的的一组地球物理勘查方法。 6.转换波:与入射波波形不同的反射波和透射波。 7.高密度电法:是集电测深和剖面法于一体的一种多装置,多极距的组合方法。 8.槽波地震勘探:是在井下煤层开采工作面内进行的,地震测线接受点和激发点沿煤巷布设,直接探测煤层内地质构造或其他地质异常体的勘探方法。 9.温纳四极装置:一种三电位电极装置,一次组合,可以获得三种电极排列的测量参数。 10.横波:质点振动方向与传播方向垂直。 11.地电断面:根据地下地质体电阻率的差异而划分界限的断面。 12.视电阻率:在电场有效作用范围内各种地质体电阻率综合反映。 13.正常时差:各观测点有不同的炮检距,因而有不同的旅行时,他们相对于自激自收时的差称为正常时差。 14.静校正:设法消除地表因素影响的校正过程。 15.观测系统:测线上激发点和接收点的相对位置关系。 16.同类波:与入射波波形相同的反射波和透射波。 17.纵波:质点振动方向与传播方向一致。 18.电测深:电测深法是根据岩石和矿石导电性的差异,在地面上不断改变供电电极和测量电极的位置,观测和研究所供直流电场在地下介质中的分布,了解测点电阻率沿深度的变化,达到测深、找矿和解决其他地质问题的目的。 19.瞬变电磁法:是利用不接地回线或电极向地下发送脉冲式一次电磁场,用线圈或接地电极观测由该脉冲电磁场感应的地下涡流产生的二次电磁场的空间和时间分布,从而来解决有关地质问题的时间域电磁法。 20.水平叠加:又称为共反射点叠加或共中心点叠加,就是把不同激发点、不同接收点上接收到的来自于同一反射点的地震记录进行叠加。 二、填空题 1.地震勘探的三个主要步骤是采集、处理、解释 2.地震勘探的横波有SV波、SH波 3.联合剖面法曲线中的正交点和反交点分别反映低阻和高阻特征 4.常用电阻率法测量方法有:电阻率测深法、电阻率剖面法、高密度电阻率法 5.观测系统图示方法有视距平面法、普通平面法、综合平面法 6.从实用性出发,地震波可分为有效波和干扰波

物探方法简介

物探方法简介 一、瞬变电磁法简介 1、瞬变电磁法技术原理 瞬变电磁法(Transient ElectromagneticsMethod, TEM)是以地壳中岩(矿)石的导电性与导磁性差异为主要物质基础,根据电磁感应原理,利用不接地回线或接地线源向地下发送一次脉冲磁场,在一次脉冲磁场的间隙期间,利用线圈或接地电极观测二次涡流场,并研究该场的空间与时间分布规律, 来寻找地下矿产资源或解决其它地质问题的一支时间域电磁法。下图即为瞬变电磁法原理的图解。 2、瞬变电磁法应用领域 瞬变电磁法施工简便、低阻探测能力强、精度高、探测深度大(地面1000m、井下150m),井下、井上均可施工。具有许多传统直流电法不可比拟的优点,可应用于: ◆地下水探测。瞬变电磁法可用于找水、咸淡水区分、地下电性

分层、圈定地下充水溶洞; ◆寻找金属矿床; ◆煤层顶底板富水性探测、巷道迎头超前探、圈定煤层采空(塌陷)区; ◆陡倾角、断层、岩脉等地质构造探测。 二、高密度电法简介 其原理与普通电阻率法相同,不同的是在观测中设置了高密度的观测点,工作装置组合实现了密点距陈列布设电极,是一种阵列勘探方法,现场测量时只需将全部电极(几十至上百根)置于测点上,然后利用程控电极转换开关和微机工程电测仪便可实现数据的快速和自动采集,增加了空间供电和采样的密度,提高了纵、横向分辨能力和工作效率。 在众多直流电阻率方法中,高密度电阻率法以其工作效率高、反映的地电信息量大、工作成本低、测量简便等突出优势,在物探领域中发挥着越来越重要的作用。主要应用于: ◆寻找地下水、管线探测、岩土工程勘察; ◆煤矿采空区调查,煤矿井下富水性探测; ◆水库大坝的坝体稳定性评价、坝基渗漏勘查、堤坝裂缝检测、建筑地基勘探; ◆涵洞和溶洞位置勘查、岩溶塌陷和地裂缝探测 三、矿井直流电法简介 主要应用于井下,其原理与地面直流电法相似,不同之处为:矿井直流电法属全空间电法勘探、采用本安防爆设备,它以岩石的电性

大学英语课文翻译

Unit1 大一新生日记 星期日 从家里出发后,我们开车开了很长一段时间才到达我住的宿舍楼。我进去登记。宿舍管理员给了我一串钥匙,并告诉了我房间号。我的房间在6楼,可电梯坏了。等我们终于找到8号房的时候,妈妈已经涨红了脸,上气不接下气。我打开门锁,我们都走了进去。 但爸爸马上就从里面钻了出来。这个房间刚刚够一个人住,一家人都进去,肯定装不下。我躺在床上,不动弹就可以碰到三面墙。 幸亏我哥哥和我的狗没一起来。 后来,爸爸妈妈就走了,只剩下我孤零零一个人。周围只有书和一个箱子。接下来我该做什么? 星期一 早上,有一个为一年级新生举办的咖啡早茶会。我见到了我的导师,他个子高高的,肩膀厚实,好像打定了主意要逗人开心。 “你是从很远的地方来的吗?”他问我。他边说话边晃悠脑袋,咖啡都洒到杯托里了。“我家离爱丁堡不太远,开车大约6个小时,”我说。 “好极了!”他说,接着又走向站在我旁边的那个女孩儿。“你是从很远的地方来的吗?”他问。但不等那女孩儿作出任何回答,他就说到,“好极了!”然后就继续向前走。他啜了一口咖啡,却惊讶地发现杯子是空的。 妈妈打来电话。她问我是不是见到了导师。 星期二 我觉得有点儿饿,这才意识到我已经两天没吃东西了。我下楼去,得知一天三餐我可以在餐厅里吃。我下到餐厅排进了长队。 “早餐吃什么?”我问前面的男生。 “不知道。我来得太晚了,吃不上早餐了。这是午餐。”

午餐是自助餐,今天的菜谱是鸡肉、米饭、土豆、沙拉、蔬菜、奶酪、酸奶和水果。前面的男生每样儿都取一些放到托盘上,付了钱,坐下来吃。 我再也不觉得饿了。 妈妈打电话来。她问我有没有好好吃饭。 星期三 早上9点钟我要去听一个讲座。我醒时已经8:45了。竟然没有人叫我起床。奇怪。 我穿好衣服,急匆匆地赶到大讲堂。我在一个睡眼惺忪的女生旁边坐下。她看了看我,问:“刚起床?”她是怎么看出来的? 讲座进行了1个小时。结束时我看了看笔记,我根本就看不懂自己写的字。 那个女生名叫苏菲,和我一样,也是英语文学专业的学生。她看起来惊人地聪明。听完讲座后我们一起闲聊。她告诉我在空档年( 高中毕业后等着上大学的一年) 里,她已经把这学期书单上的书都读完了。她给我留下了深刻的印象,我觉得自己太无知了,甚至不配跟她呼吸同样的空气。 妈妈来电话问我睡得好不好。 星期四 今天是新生集会(社团招新活动)。我和苏菲跑去看看我们能加入几个俱乐部。我们俩都认为我们应该结交很多朋友,所以我报名参加了交谊舞俱乐部、人工智能协会、手铃俱乐部和极限运动俱乐部。苏菲则报名参加了业余剧社和莫扎特合唱团。 我不知道我和苏菲还能不能继续做好朋友。 妈妈来电话。她告诉我哥哥曾试图把我在家住的那间卧室租出去。妈妈向我保证只要我需要,那房间永远是我的。她还说那是我的家,他们都非常想我,尤其是我的狗。我一下子就哭了起来。 星期五 早上我去了图书馆。可是好像我需要一个能验明我身份的证件才能进图书馆。不知为什么,我必须发誓不损坏书籍、不违反图书馆的规定,否则我就要被当作罪犯被送进监狱。(什么!?就因为说话声音太大?)图书馆看起来很古老,学校以此为豪。 今晚有个迪斯科舞会,可我已经没有干净衣服穿了。我只知道把脏衣服扔进衣篮中,但并不清楚衣服是如何洗净,熨平并叠好放进衣柜里的。也许妈妈快来电话了。

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