kohn  sham ( ks ) 
 density functional theory ( dft ) provides 
 a powerful theoretical framework for performing calculations on a 
 subsystem exactly embedded in its full , quantum - mechanical environment . the quantity that mediates the interaction between subsystems is 
 the electronic density , which is partitioned into two terms the ks energy consists of terms relating 
 to subsystems a and b 
 and an interaction term containing all nonadditive parts of the energy : this last term includes an explicitly known 
 coulomb contributionand , depending on the details of the subsystem 
 partitioning , simple electrostatic interaction of electrons in a with 
 nuclei in b and vice versa . exchange - correlation 
 effects between subsystems are included throughwhich of course must be approximated in practical 
 calculations . if the subsystem densities a and b are constructed from mutually orthogonal 
 orbitals , the kinetic 
 energy of the whole system is simply given by the sumin general , however , there will also be a 
 nonadditive termwhich must be approximated or , 
 in more recent versions of the theory , computed exactly using optimized 
 effective potential ( oep ) methods . the current 
 authors have demonstrated that oep methods can be used 
 in an iterative scheme to obtain the kohn sham ground state 
 density for molecular systems , including those that exhibit strongly 
 overlapping subsystem densities . reiher and co - workers 
 used oep methods to analyze and compare accurate embedding potentials 
 in approximate embedding schemes , and 
 oep methods were used to solve for a unique partitioning of a system 
 by allowing the subsystems to share a common embedding potential . finally , several groups have utilized these dft embedding methods 
 to describe the interface between high - level and low - level subsystems 
 in wave - function - in - dft ( wf - in - dft ) approaches . in what follows , we avoid the complications 
 of oep through three 
 simple , robust innovations : ( 1 ) we replace the iterated dft - in - dft 
 with a single conventional ks calculation . ( 2 ) we completely avoid 
 the issue of kinetic energy nonadditivity through the use of a level 
 shifting projection operator to keep the orbitals of one subsystem 
 orthogonal to those of another . and , ( 3 ) we develop a simple but effective 
 perturbation theory to eliminate practically all dependence on the 
 level shift parameter . embedding methods that maintain orthogonality 
 between subsystem 
 orbitals have been in use for decades . what has not been recognized 
 is that these can be used to formulate a formally exact dft embedding 
 scheme , equivalent to but much simpler than the recently developed 
 oep - based methods . the philips kleinman pseudopotential approach 
 starts by level shifting the core orbitals to produce valence orbitals implicitly orthogonalized to the core . even frozen core approximations , which have been in use in quantum 
 chemistry at least since the 1950s , amount 
 to a sort of hartree  fock ( hf)-based embedding scheme in which 
 the core and valence subsystems are described by mutually orthogonal 
 orbitals . similar ideas operate in the incremental scheme introduced 
 by stoll and co - workers , in the region 
 method of mata and et al . , and in henderson s 
 embedding scheme . further details of 
 the large amount of literature in this area can be found in a recent 
 and comprehensive review . the pauli 
 principle is enforced through a level - shift projector 
 in the ab initio model potential ( aimp ) method , which in turn derives from earlier work by huzinaga and cantu . a similar strategy was employed by rajchel et 
 al . for the description of intermolecular interactions , but there 
 they iterated equations for each subsystem . projection is sometimes employed in the embedding used in the fragment 
 molecular orbital method ( see for example the recent article ref ( 23 ) and references therein ) . the novelty of the current work resides in the important simplification 
 that arises by performing a conventional kohn  sham calculation 
 plus localization to determine fragment orbitals , the recognition 
 that this amounts to a formally exact dft - in - dft embedding scheme , 
 and the elimination of practically all dependence on the level - shift 
 parameter through perturbation theory ( see below ) . self - consistently 
 iterated embedded dft calculations are exactly 
 equivalent to full ks theory on the whole system . if the only aim 
 is to perform dft - in - dft embedding , it is currently more practical 
 to perform ks theory on the whole system . we now demonstrate that 
 this remains true even if the initial mean - field calculation is merely 
 a prelude for wf - in - dft embedding . the sum of the atomic - orbital 
 ( ao ) density matrices from the embedded 
 one - electron equations is equal to the total ks density matrix :  +  = . 
 the occupied ks orbitals { i } can 
 be rotated among each other , and one rotation produces the subsystem 
 orbitals { ia }  { ib}. let us assume that 
 the whole system and both subsystems a and b are closed - shell and 
 that the total number of electrons is partitioned as na + nb = n ; the extension of the following arguments to open - shell subsystems 
 is straightforward . to produce the correct total electron density , 
 a ks calculation 
 on subsystem a requires an embedding potential that correctly describes 
 the electrostatic and exchange - correlation interactions with the electrons 
 in subsystem b , while also properly enforcing pauli exclusion between 
 the subsystem densities . this is in fact the role of the nonadditive 
 kinetic potential vnad , but an alternative scheme can be reached by simply level 
 shifting the orbitals of b to very high energy for the calculation 
 on a. then , a correctly embedded calculation on the na electrons in subsystem a has the energy expression1where h is the core hamiltonian 
 ( kinetic plus external potential ) in the ao basis . we propose 
 a fock operator for electrons in subsystem a that includes 
 the appropriate derivative of the energy , plus a level - shift term 
 to preserve the orthogonality relation ia|jb = 0:2wherethe term p elevates the energy of the ith orbital in subsystem 
 b to ib +  , and provided that  is sufficiently 
 large and positive , the eigenstates of f   are then mutually orthogonal to the orbitals in subsystem 
 b. evaluation of the gradient in eq 2 produces 
 the fock operator for subsystem a that we employ in the calculations 
 below : in hf theory or in ks theory with hybrid functionals , 
 the additional ( possibly scaled ) exchange term k[ +  ] appears in the usual 
 way . as in other dft - in - dft embedding schemes , this approach 
 provides 
 the flexibility to perform a completely different type of electronic 
 structure calculation for subsystem a , using a core hamiltonian that 
 contains the additional embedding terms from above:3the total energy for a normalized wave function 
 in the a subsystem is then calculated aswhere  is the hamiltonian built with the embedding potential added to the 
 core hamiltonian , the central terms correct the exchange - correlation 
 energy contribution , and the final term is the energy of the electrons 
 in the b subsystem . the code is implemented in the development 
 version of the molpro 
 software package . the current version of the code , 
 and all calculations presented here , employ the supermolecular basis 
 set . future versions will employ subsystem calculations in a spatially 
 local set of atomic orbitals ; we see no technical barriers to this 
 extension . we begin by demonstrating the efficacy of the projection technique 
 described above , using the example of embedding the 10 electrons of 
 the oh moiety of ethanol in the environment produced by the 
 ethyl subsystem . ks orbitals from a pbe calculation in the 6 - 31 g * basis are localized 
 using the pipek mezey localization scheme , and the five orbitals clearly associated with the oh 
 group are identified . note that this step is performed automatically by finding those orbitals with a mulliken population on any  active the embedded core hamiltonian matrix h is then constructed according 
 to eq 3 , with  = 10eh , and a 10-electron kohn  sham calculation 
 is performed . if the projection technique has functioned perfectly , 
 the resulting orbitals will be orthogonal to those associated with 
 subsystem b , the energy expression in eq 1 will 
 yield precisely the ks energy of the whole system , and  tr 
 p will exactly vanish . the value of  tr p is approximately 20 eh , and the energy is somewhat too low . in the limit of infinite  , 
 the agreement with kohn sham theory will be exact by construction , 
 so the error arises from the finite value of this parameter . upon 
 increasing the value of  , we do indeed find that the error 
 in the energy decreases ( figure 1 , uncorrected ) , 
 although very large values result in numerical noise associated with 
 the double - precision numerics of the program . error in the uncorrected 
 ( e[; ] ) and 
 corrected ( e[; ] +  tr p ) energy 
 expressions relative to full ks on ethanol 
 using pbe/6 - 31 g * , demonstrating that the perturbation theory correction 
 yields essentially exact embedding energies over a wide range of . a more stable approach is to compute the   
  
 limit using perturbation theory . we can introduce a perturbed operator f that satisfies the requirements f0 = f + p andby adopting the form(here , for clarity , we have dropped the a 
 and b superscripts . ) expanding the operator f as a power series in  and making the usual 
 perturbative expansion of energy and wave function , we obtainwhich yields at first - order the contribution e1 = 0|p|0. in the ao basis e1 =  
 tr p , and as shown in 
 table 1 , this correction brings the embedded 
 energy into near - perfect agreement with the ks energy . higher orders 
 in this perturbation expansion could be used to correct the embedded 
 orbitals , or to further correct the energy , although nothing in our 
 calculations so far suggests that this will be necessary . in 
 fact , it turns out that the energy expression e[; ] +  tr p agrees with the full ks energy 
 to 7 peh , which is essentially exact in 
 double precision . this excellent agreement is very insensitive to 
 the choice of  , as shown in figure 1 . for values < 10eh , the electronic structure is qualitatively incorrect , and for  
 > 10eh numerical noise 
 begins 
 to become an issue . but for all values of  in between , spanning 
 5 orders of magnitude , the discrepancy from full ks theory is below 
 20 neh . similar results are found 
 for other molecules , and the projection 
 approach can also be employed for hf embedding . for example , a hf - in - hf 
 embedded calculation on pyradine with subsystem a composed of the 
 five pipek mezey orbitals associated with the nitrogen atom 
 and with  = 10eh reproduces 
 the total hf energy to within 0.1 neh . the method is also very stable for wf - in - dft calculations , which we 
 will demonstrate through ccsd(t)-in - pbe calculations on ethanol , using 
 the same subsystem partitioning as before . the calculations are performed 
 ( i ) without the perturbative correction to the total energy , ( ii ) 
 with the pertubative correction obtained using the hf subsystem density 
 for the active subsystem , and ( iii ) with the perturbative correction 
 obtained using the unrelaxed ccsd density for the active subsystem ; 
 the differences between the total energies with  = 10 and  = 10 are then 470 neh , 50 neh , and 0.74 neh , respectively . a key motivation 
 for dft embedding schemes is the possibility of embedding accurate 
 wave function calculations in a dft environment . here , we demonstrate 
 the simplicity and robustness of the current scheme with a few , sample 
 calculations . first , we consider the deprotonation energy of 
 gas - phase ethanol , an example for which the pbe / aug - cc - pvdz level 
 of theory is in fairly good agreement with ccsd(t)/aug - cc - pvdz . two 
 embedded calculations are performed , using either oh ( 10 electrons ) 
 or ch2oh ( 18 electrons ) as the active subsystem . the results are shown in table 2 , where it 
 can be seen that embedded ccsd(t)-in - pbe with the 10-electron active 
 subsystem slightly overcorrects the pbe value , by approximately 6 
 meh . the calculation with the 18-electron 
 active subsystem produces a result in much better agreement , differing 
 from the total ccsd(t ) energy by a little over 1 meh . these results are not entirely unexpected , since in 
 the first calculation the system is partitioned across a bond which 
 changes polarization during the deprotonation process . the reactant and transition 
 state structure were optimized at the b3lyp/6 - 311g*++ level of theory using gaussian . three coupled - cluster , 
 ks , and embedded activation energies are shown in table 3 . activation energies calculated at the ccsd(t ) , dft , and 
 ccsd(t)-in - dft levels of theory are shown in table 3 ; each of these calculations is performed using the cc - pvtz basis set with aug - cc - pv(t+d)z on chlorine . dft calculations with approximate functionals 
 often underestimate barriers , but in the present case , the pbe and 
 blyp barriers are actually negative . the embedded results are consistently much closer to the reference 
 coupled - cluster results , with the accuracy improving as more atoms 
 are included in subsystem a. the basis set is cc - pvtz with 
 aug - cc - pv(t+d)z on chlorine . recent research in the group of one of us has focused on the use 
 of embedded many - body expansions to compute accurate structure and 
 energetics of molecular crystals . the 
 idea builds on earlier attempts to capture many - body effects through 
 embedding , and it offers an important step toward systematically 
 improving accuracy by improving the embedding scheme , rather than 
 the number of terms in the many - body expansion . we illustrate this 
 approach for the case of a symmetric model water trimer . for 
 simplicity , the trimer is optimized in a planar geometry , in which 
 each of the three monomers have equivalent geometries and environment 
 ( see structure in figure 2 ) . the geometry is 
 optimized with these contraints using mp2/6 - 31g*. denoting the monomer , 
 dimer , and trimer ccsd(t ) energies as e1cc , e2cc , and e3cc , we can immediately express the coupled - cluster binding energy ( relative 
 to water molecules in their cluster geometry ) in the form ebindcc = e3cc  3e1cc . binding curve relative to monomers for a planar 
 water trimer as 
 a function of the distance from the center of mass to the oxygen atoms . 
 shown are the binding curves from hf theory , ccsd(t ) many - body expansion 
 truncated at the two - body level ( mbe2 ) , embedded two - body expansion 
 ( embe2 ) , and from full trimer ccsd(t ) . in the lower panel , the error for 
 embe2 is compared to that of the simple additive scheme hf-12 , in which the hf binding curve is corrected using the two - body 
 ccsd(t ) correlation energies . several important approximations can be defined 
 without embedding , 
 and two of these are shown in figure 2 . truncating 
 the many - body expansion at the two - body level gives the approximation4where e2cc = e2cc the hf approximation to the binding energy is defined by ebindhf = e3hf  3e1hf . both of these approximations result 
 in errors of around 5 meh in the total 
 binding energy of around 20 meh . since 
 our hf - in - hf embedding calculations require hf on the full trimer , 
 we focus our attention on the correlation contribution to binding , 
 and we use the notation e  e the one - body correlation correction without 
 embedding vanishes ( by analogy to eq 4 ) , but 
 with embedding , the contribution added to the trimer energy is different 
 from that subtracted as part of the monomer energy . thus , one obtains 
 the approximationwhere 1corr is the correlation 
 energy from a cc - in - hf calculation on a single monomer . a further 
 correction can be made by considering the two - body embedded correlation 
 contribution , givingwhere 2corr this embe2 approximation can be 
 seen in figure 2 to be in essentially perfect 
 agreement with ccsd(t ) across 
 all geometries . also shown in the figure is the error in this scheme 
 ( compared to ebindcc ) , which can be seen to be on the order of 
 10 eh in the binding region . the 
 results are strikingly better than those obtained using the simple 
 unembedded hf-12 approach defined asebindhf-12 = ebindhf + 3e2corr , especially at short range . the ideas presented here 
 derive from a long history of ensuring 
 that the pauli principle is obeyed through level shifting a subset 
 of the orbitals . but the interesting and novel point is that the dft 
 embedding techniques recently developed by us and 
 by others can be cast exactly into the very simple and robust framework described 
 here . it should be noted that this projector embedding method 
 is not 
 without drawbacks . it is limited to applications for which the electronic 
 structure can be reasonably described using ks theory , because the 
 projector can only be formed if ks orbitals are available . this limitation 
 will become problematic in cases for which the dft description is 
 fundamentally broken , or if it is impractical to perform a full ks 
 calculation on the entire system . furthermore , the new method describes 
 embedding in the environment by a projection operator , rather than 
 a simple , local potential . in its favor , however , the current 
 approach relies on existing , 
 stable , well - developed technologies . it requires nothing more than 
 a ks code , an orbital localization scheme , some elementary matrix 
 operations in the atomic orbital basis , and any wave function method 
 that can accept an arbitrary core hamiltonian . since these ingredients 
 are available in practically all molecular electronic structure codes , 
 we anticipate no barriers to widespread adoption of the approach .