the retinal protonated 
 schiff base ( rpsb , represented in figure 1a ) 
 is the chromophore responsible for the photochemical 
 properties of a vast family of biological photoreceptors referred 
 to as retinal proteins which are , among other functions , involved 
 in the mechanism of vision of dim light in vertebrates . trans isomerization in the protein ( opsin ) environment ( 200 fs ) 
 with high quantum yield ( 0.68 ) upon photon absorption . this isomerization process has been deeply investigated 
 by different spectroscopic techniques and 
 theoretical calculations . in particular , femtosecond 
 spectroscopy and hybrid quantum mechanics / molecular 
 mechanics ( qm / mm ) molecular dynamics calculations highlight 
 the essential role played by the surrounding protein environment and 
 by the hydrogen - out - of - plane motion in 
 the isomerization mechanism , which involves decay via a s1/s0 conical intersection ( ci ) reached by the selective torsion of the central ( c11=c12 ) double bond . ( a ) lewis representation of the full chromophore , 
 the 11-cis retinal protonated schiff base ( rpsb ) , 
 covalently bound 
 to a lysine of the opsin protein , and of the cis-,trans - penta-2,4-dieniminium cations ( psb3 ) . ( b ) schematic 
 representation of the thermal isomerization from the cis to the trans configuration of psb3 , following the 
 two minimum energy paths ( obtained in ref ( 11 ) from casscf calculations ) undergoing respectively 
 a charge transfer ( ct ) and a diradical ( dir ) transition state . the 
 former minimum energy path ( mepct ) is represented in red with a saddle 
 point in the charge transfer transition state ( tsct ) , and the orange 
 region indicates that the ground state potential energy surface is 
 characterized by a charge transfer character . the latter minimum energy 
 path ( mepdir ) is represented in blue with a saddle point in the diradical 
 transition state ( tsdir ) , and the cyan region indicates that the ground 
 state potential energy surface is characterized by a diradical character . the energy barriers corresponding to tsct and tsdir are comparable , 
 and most of the computational studies agree that tsct represents the 
 lowest barrier . the main difference between tsct and tsdir configurations 
 lies in their bond length alternation ( bla ) , which is positive for 
 the tsdir and negative for the tsct ( see discussion in the text ) . a scan of the configurations connecting tsct with tsdir ( called bla 
 scan ) presents a conical intersection ( ci ) for a positive bla , but 
 the actual value varies with the adopted computational method . the penta-2,4-dieniminium cation 
 ( psb3 , represented in figure 1a ) is a small 
 conjugated molecular system widely 
 used as reduced computational model of the full rpsb because psb3 
 and rpsb exhibit similar ground and excited state features . first , 
 both psb3 and rpsb have equilibrium cis and trans isomers where the positive charge is localized on 
 the nitrogen - containing side of the conjugated chain . second , the 
 transition from the ground state ( s0 ) to the first excited 
 state ( s1 ) is characterized by a transfer of the positive 
 charge from the nitrogen region toward the opposite end of the conjugated 
 chain . third , twisting the conjugated chain along one of the central 
 double bonds leads to a ci structure between the s0 and 
 s1 electronic states . such a ci plays an important role 
 in the photoisomerization mechanism of rpsb because it mediates population 
 transfer from s1 to s0 along the isomerization 
 coordinate . not only do psb3 and rpsb have similar near - equilibrium 
 and s1 potential energy surface features , but it was recently 
 found 
 that they have similar ground state potential energy features in the 
 vicinity of the ci as well . in a hybrid quantum mechanics / molecular 
 mechanics ( qm / mm ) study of the thermal isomerization mechanism of 
 bovine rhodopsin , it was found that thermal 11-cis to all - trans isomerization of rpsb may occur via 
 one of two distinct saddle points that are in the vicinity of the 
 s0/s1 ci . both saddle 
 points ( i.e. , chemically , transition states ) feature an almost orthogonally 
 twisted c11=c12 double bond , similar 
 to the ci but very different bond length alternations ( bla ) . we define 
 the bla as the difference between the average bond length of formal 
 single bonds ( c1c2 and c3c4 for psb3 ) and the average bond length of formal 
 double bonds ( c1=n , c2=c3 , and c4=c5 for psb3 ) , such that bla 
 is positive if there is no bond inversion and negative if there is . indeed , psb3 also 
 features two transition states which are almost 90 twisted along 
 the central ( c2=c3 ) double bond but having 
 different bla patterns . in both psb3 and rpsb , the two transition 
 states not only have different bla geometries but also different electronic 
 structures . one transition state ( tsct ) is characterized by a transfer 
 of the positive charge from the nitrogen - containing side of the molecule 
 to the other end of the conjugated chain . the other transition state 
 ( tsdir ) retains the positive charge on the nitrogen side of the chain 
 and therefore has covalent / diradical character due to the homolytic 
 breaking of the isomerizing double bond . to characterize the regions 
 of the ground state potential energy surface driving the thermal isomerization , 
 gozem et al . optimized the two transition 
 states in psb3 and used them to map three pathways at the casscf/6 - 31 g * 
 level of theory . the first two pathways are minimum energy paths ( meps ) 
 leading away from each transition state and toward the two equilibrium 
 structures , cis - psb3 and trans - psb3 . these paths are called mepct ( for the path passing through the charge 
 transfer transition state tsct ) and mepdir ( for the path passing through 
 the covalent / diradical transition state tsdir ) . the third path is 
 an interpolation / extrapolation of coordinates between the two transition 
 states . because the transition states essentially have different blas 
 but otherwise are almost geometrically identical , this scan follows 
 a bla coordinate and is therefore called the bla scan . this scan intercepts 
 a s1/s0 ci structure . for a simplified representation 
 of the three paths , the three aforementioned paths served 
 useful as benchmarks to understand 
 the importance of a balanced representation of electronic correlation 
 in correctly describing the topology of the potential energy surface 
 in that region as well as in computing the relative energies of tsct 
 and tsdir . if the casscf(6,6)/6 - 31 g * can be considered a good compromise 
 between accuracy and computational effort for mapping ( small ) parts 
 of the potential energy surface of complex molecular systems , using 
 more sophisticated methodologies for single - point energy calculations 
 on the casscf paths is seen to be a reliable strategy to explicitly 
 include dynamical correlation in the wave function , allowing one to correct the original casscf findings 
 such as ci location , energy of the  and  * 
 configurations , and relative stability of the tsct and tsdir . thanks to the tremendous progress in high performance computing 
 ( hpc ) in recent years , quantum monte carlo ( qmc ) methods represent 
 a powerful alternative to other ab initio and dft approaches for the 
 accurate description of systems where electronic correlation plays 
 an essential role . qmc methods have been widely applied 
 to several problems of physical and chemical interest such as materials , molecular properties , reaction pathways , and geometry and excited states 
 of biochromophores . the good scalability with respect to the system size ( n , with 3 < d < 
 4 and n the number of electrons ) and the use of massively parallel algorithms make qmc methods particularly 
 suitable for petascale architectures . all these aspects justify the 
 growing number of applications of qmc in problems of quantum chemistry 
 and molecular physics . the variational monte carlo ( vmc ) method is the simplest qmc approach : thanks to a combined 
 use of the monte 
 carlo integration and the variational principle for the ground state , 
 the many - body trial wave function t can be efficiently optimized and used to extract information other 
 than just the energy of the molecular target . higher accuracy in the 
 determination of the system properties ( energy , geometrical and electronic 
 parameters , etc . ) is usually achieved by the fixed node projection 
 monte carlo methods such as the diffusion monte carlo ( dmc ) or the lattice regularized diffusion monte carlo ( lrdmc ) . the choice of the functional form of t represents a crucial step determining the 
 overall quality 
 of qmc calculations , both in the variational and in the fixed node 
 projection schemes . the jastrow antisymmetrized geminal power ( jagp ) has been seen to be very efficient in the investigation of chemical 
 systems , with accuracy comparable to that of high - level quantum 
 chemistry methods . its compactness , coupled to the use of efficient 
 algorithms for the optimization of all parameters , including linear 
 coefficients and exponents of the atomic basis set , leads to a fast convergence of the variational 
 results for electronic and geometrical properties with the size of 
 the basis set , with a computational cost comparable to that of a simple wave function 
 defined by the product of a jastrow factor and a single slater determinant . although agp is not in general size consistent , the presence of a 
 flexible jastrow factor makes jagp size consistent for systems which 
 splits in fragments with spin zero or 1/2 . jagp has already proven to give a good description 
 of the static and dynamical correlation in several crucial benchmarks like in the estimation of the torsional energy of the ethylene and 
 of the singlet  triplet gap of methylene . in the following , we compute the energy and electronic 
 structure 
 of psb3 along the mepct , mepdir , and bla paths reported previously . as shown below , the combination of qmc and jagp 
 ansatz allows us to get a proper description of both static and dynamical 
 correlation ; for this reason , using the qmc and jagp wave functions 
 represents an optimal choice to study the intrinsic properties of 
 the psb3 model . the paper is organized as follows : in section 2 we report the main features of vmc and lrdmc schemes 
 and 
 a detailed analysis of the jagp ansatz and of its potential to correctly 
 describe multiconfigurational systems such as diradicals ; computational 
 details on our calculations are given in section 3 ; the current results are shown in section 4 , pointing out the importance of the electronic correlation 
 properly introduced by the qmc / jagp study for the characterization 
 of the conical intersection ; conclusions and comments on future perspectives 
 are reported in the last section . the accuracy of qmc approaches , both in the simplest 
 vmc scheme and in the fixed - node projection schemes , are strictly 
 related to the wave function ansatz . typically , the electronic wave 
 function t in qmc is defined by the product1where   is the antisymmetric 
 function taking into 
 account the fermionic nature of electrons and   is the 
 jastrow factor depending explicitly 
 on the interparticle ( electrons and nuclei ) distances ; x and r represent the collective electronic ( x refers to space r and spin  ) 
 and nuclear coordinates , respectively . the jastrow factor is 
 a symmetric positive function of the electronic positions ; therefore 
 it does not change the nodal surface ( determined by the antisymmetric 
 term ) , but it introduces 
 the dynamical correlation 
 among electrons and satisfies the electron  electron and electron the parameters that define t are optimized in order to minimize the electronic energy within 
 the functional freedom of the ansatz . the vmc results can further 
 be improved by using the fixed - node ( fn ) projection monte carlo techniques , 
 which provide the lowest possible energy with the constraint that 
 the wave function fn has the same nodal surface 
 of an appropriately chosen guiding function t , which is usually optimized using the vmc 
 method . the fixed - node projection monte carlo method that we have 
 adopted is lrdmc , which is efficient for systems 
 with a large number of electrons and 
 preserves the variational principle even when used in combination 
 with nonlocal pseudopotentials . because 
 the lrdmc calculations are much more demanding than the vmc calculations , 
 in terms of computational time , they have been performed only for 
 a few key structures . the trial wave function ansatz used in the 
 qmc calculations presented 
 in this paper is the jastrow antisymmetrized geminal power ( jagp ) , that is , the product2of 
 the antisymmetrized geminal power ( agp ) 
 function agp and the jastrow factor j , where the dependence on the nuclear coordinates r is here omitted . for an unpolarized system ( zero total spin s ) of n = 2np electrons and m atoms , the agp function is defined 
 as3where   is 
 the antisymmetrization operator and 
 the geminal pairing function g is a product of a 
 singlet function and a spatial wave function symmetric with respect 
 to the particle exchange : 4the 
 spatial function   is a linear 
 combination of products of 
 atomic orbitals :5where the indexes  and  run 
 over all the basis in all the atoms in the system , for a total of l atomic orbitals ( note that l is determined 
 by the overall basis set size ) . the coefficients g have to be optimized in order to minimize 
 the variational energy of the system ( together with the other parameters 
 in the wave function ) . in our calculations , we used this jastrow 
 factorthat involves 
 the one - electron interaction 
 term uen , the homogeneous two - electron 
 interaction term uee , and the inhomogeneous 
 two - electron interaction terms ueen and ueenn ( representing respectively an electron  electron  nucleus 
 function and an electron they are defined as follows:6789where 
 the vector ria = ri  ra is the difference between the 
 position of the nucleus a and the electron i , ria is the 
 corresponding distance , rij is the distance between electrons i and j , za is the 
 electronic charge of the nucleus a , which is described 
 by la atomic orbitals 
 a ( with index  = 1, ... ,la ) , and b1 , b2 , faa , f,a , and f,a , b are variational parameters . [ note that the atomic 
 orbitals  used in the jastrow term are similar to the atomic 
 orbitals  used for the agp , although they are not the same 
 orbitals , and in general a reliable description of molecular systems 
 requires a much smaller number of orbitals in the jastrow than the 
 number of orbitals used in the agp , see for instance ref ( 56 ) . ] the leading contribution 
 for the description of electronic correlation is given by uee , but also the inhomogeneous two - electron 
 interaction terms ueen and ueenn are particularly important in the jagp ansatz because 
 they reduce the unphysical charge fluctuations included in the agp 
 function , as discussed in refs ( 25 , 49 ) . the pairing spatial function   in eq 5 is written 
 in terms of the ( localized ) atomic orbitals  , offering an interesting correspondence between the agp ansatz and 
 the resonating valence bond framework . an equivalent 
 way to write the pairing function   is obtained 
 by using the molecular orbitals 
 ( mos ) k . the expansion of the pairing 
 function in terms of mos is obtained by performing a generalized ( the 
 atomic orbitals  are not necessarily orthonormal , 
 so the overlap matrix s = 
 | 
   ) diagonalization of the 
 coupling matrix g , which is the l  l matrix of the g coefficients:10where  = diag(1, ... ,l ) and11the resulting 
 pairing function   is12where the 
 orthonormal single particle functions 
 are written as13with the pk coefficients defining the eigenvectors p. it is interesting to investigate the connection between the 
 expansion 
 of the pairing function   in terms of 
 mos and the standard configuration 
 interaction expansion of the wave function in multiconfigurational 
 approaches . by substitution of eq 12 in 
 eq 3 , and expanding the summation out of the 
 antisymmetrization 
 operator , the following multideterminant expansion is obtained for 
 the agp function:14where the coefficients are 
 given by15|0 is the leading 
 closed - shell slater determinant:16the determinant |iiaa is equal to |0 , 
 but with 
 the virtual orbital a substituting 
 the valence orbital i , etc . from 
 the expression of the coefficients in eq 15 and 
 the ordering of the eigenvalues k in eq 11 , it follows that the leading contribution 
 beyond the determinant |0 is given by the 
 determinant |iiaa with i = np and a = np + 1 . the multideterminant expansion 
 of agp in eq 14 allows us 
 to directly compare the agp with wave functions 
 from other quantum chemical frameworks . in agp , all the odd excited determinants ( single , triple , etc . ) are excluded , 
 whereas a subset of the even excitations ( those with a multiple excitation 
 to the same virtual orbital ) are taken into account ; only doubly occupied 
 molecular orbitals are present . in other words , agp is contained in the seniority zero sector of the electronic full 
 configuration interaction and its expansion coefficients are determined 
 by the ratios of the eigenvalues of the  matrix . the 
 seniority number  represents an alternative tool to 
 classify singlet wave functions . is defined as the number 
 of unpaired electrons in the slater determinant , i.e. , the number 
 of singly occupied molecular orbitals . in cases when the static correlation 
 plays a major role -based selection of important slater determinants 
 in the expansion has been seen to be superior than the traditional 
 one , based on the number of excitations with respect to the reference 
 configuration . wave functions with  
 = 0 for benchmark systems are accurate enough to recover most of the 
 static correlation , but the fci limit ( including dynamical correlation ) 
 is achieved only when configurations from  = 2 , 4 , 6 , ... sectors 
 are explicitly included . in the case 
 of jagp wave function , the combination between a  = 0 determinantal 
 term and a jastrow factor allows us to estimate the correlation energy 
 more accurately than  = 0 ci wave functions . the set of mos 
 k is optimized within the jagp 
 framework , i.e. , in the presence of the jastrow factor and of the 
 multiconfigurational character of the wave function : mos extracted 
 from our optimization procedure represent therefore the optimal choice 
 for the correlated description of the system under study . the way 
 to move from the agp mos to the standard ones and vice versa is explained 
 in the next paragraph and in appendix . the multiconfigurational 
 nature of the agp function , clearly shown by eq 14 , has been extensively discussed by zen et al . in a recent paper 
 on the use of agp for diradicals . here we focus the attention on the application of the agp ansatz to the 
  * state of psb3 ( see the discussion in section 4 ) , starting from the simple but accurate model with 
 two electrons in two orbitals , originally introduced by salem and 
 rowland . for a generic diradical system , two atomic 
 orbitals a and b are centered 
 on nuclei a and b. such a model is a representative scheme for molecules 
 undergoing a double bond breaking , like the twisted ethylene or the 
 twisted psb3 : in these cases , the two involved orbitals are of p type and positioned on two ( central ) carbon atoms . the main goal of this discussion is to demonstrate that the agp ansatz , 
 a formally  = 0 wave function , contains the  * 
 configuration , needed for a proper description of the ground state 
 surface of psb3 along the paths . [ the  and  * orbitals 
 have to be intended here as -molecular orbitals residing on 
 highly twisted configurations . ] in case of two orbitals , the 
 pairing function   term in the 
 agp formulation is explicitly 
 given by17where g coefficients represent 
 the coupling terms of the g matrix 
 in the expansion of the agp spatial factor : gaa and gbb are referred to the 
 ionic terms a(r1)a(r2 ) and b(r1)b(r2 ) in which the two electrons are localized on the same atom , whereas 
 the elements gab and gba are related to the covalent terms a(r1)b(r2 ) and b(r1)a(r2 ) . in terms of molecular 
 orbitals ,   becomes ( l = 2)18on the other hand , in the standard delocalized 
 picture two + and  molecular orbitals are defined:19assuming zero overlap 
 between a and b ( the assumption 
 is easily verified 
 for two orthogonal p orbitals ) . the diradical 
 (r1,r2 ) = + wave 
 function ( + is a short 
 notation for +(r1)(r2 ) + (r1)+(r2 ) ) is written in the atomic basis as the following20because the agp formally contains the terms 
 introduced in eq 20 , the comprehension of a 
 consistent way to link together eqs 17 , 18 , and 20 is a mandatory task . 
 some questions are therefore arising : ( i ) which relation occurs between 
 1 , 2 , + , and 
  ; ( ii ) which relation exists between the 
 atomic basis , a and b , and the 
 molecular orbitals , 1 and 2 obtained 
 by the diagonalization of the   matrix ( eq 18 ) ; ( iii ) 
 finally , if a diradical ground state + can be properly described by the agp ansatz . a unitary matrix   transforms the + and 
  molecular orbitals , by rotating them with 
 a certain angle  , into + and 21if 
  = /4 , the normalized orbitals 
 become2223by defining 1  + and 2  
  , which become 1 = a and 2 = b by taking into account the relations 
 in eqs 19 , and by substitution in eq 18 , we obtain24which is equivalent to eq 20 for 1 = 2 = 1/2 . we also observe that eq 17 is equivalent to 
 eq 20 for gab = gba = 0 and gaa = 
 gbb = 1/2 . summarizing , 
 it is always possible to transform a + configuration into a combination of +2 and 2 by applying 
 an opportunely chosen proper unitary transformation . in this simple 
 model for diradicals , the agp molecular orbitals are a linear combination 
 of the orbitals deriving from the traditional picture of electronic 
 delocalization . the detailed derivation of the mapping procedure is 
 reported in the appendix , where the generalization 
 from the ( 2,2 ) to the ( 2,n ) active space ( where n is the number of orbitals ) is shown and the proof of the 
 capability of agp to represent such active space is reported . the 
 expansion in eq 17 shows that the agp ansatz 
 contains all the terms reported by the picture in terms of delocalized 
 molecular orbitals and localized atomic orbitals . the g coefficients are variational parameters 
 optimized by the stochastic methods mentioned before , and for this 
 reason the agp optimization is a fundamental step to select the right 
 wave function for the ground state of interest . as a conclusion 
 of this discussion , the graphical representation 
 of casscf and agp molecular orbitals for highly twisted diradical 
  * state of a certain structure of psb3 ( the last structure 
 of the bla path , see the discussion below ) , together with the combinations 
 explained in this paragraph , is reported in figure 2 : the agp frontier orbitals ( 1 and 2 , eq 18 ) are identical to the normalized 
 sum and difference of the corresponding casscf orbitals (  and 
  * orbitals ) , according to the rotation given by the unitary 
 transformation . comparison between casscf frontier (  
 and  * ) and agp 
 frontier ( 1 and 2 , see eq 18 ) orbitals ; corresponding combinations , given the 
 unitary transformation   ( eq 21 ) , are also 
 shown . the qmc calculations reported in this paper have 
 been obtained 
 using the turborvb package , developed by s. sorella and co - workers , that includes a complete suite of variational 
 and diffusion monte carlo codes for wave function and geometry optimization 
 of molecules and solids . the scalar - relativistic energy consistent 
 pseudopotentials ( ecp ) of burkatzki et al . have been used in order to describe the two core electrons of the 
 carbon and nitrogen atoms . in detail , the basis sets we have used 
 for the agp part are ( 10s,9p,2d,1f ) contracted in { 8 } hybrid orbitals 
 for the carbon atom , ( 8s,9p,2d,1f ) contracted in { 8 } hybrid orbitals 
 for the nitrogen atom , and ( 6s,5p,1d ) contracted in { 1 } hybrid orbitals 
 for the hydrogen atom . as basis sets for the atomic orbitals included 
 in the inhomogeneous terms of the jastrow factor , namely in uen , ueen and ueenn reported in eqs 6 , 8 and 9 , we used an 
 uncontracted basis for the uen and ueen term and a contracted with hybrid orbitals 
 basis for ueenn . this allowed us to have 
 an accurate basis set for the jastrow factor while keeping the number 
 of parameters of the wave function reasonably small . in more details , 
 in uen and ueen we used a ( 4 s,2p,1d ) basis set for the carbon or nitrogen atoms , 
 and a ( 3 s,2p ) for hydrogen atom , whereas in ueenn the orbitals are contracted in { 2 } hybrid orbitals for 
 the oxygen , nitrogen , or hydrogen atoms . the parameters of the 
 wave function t , including the 
 values of the exponents of the atomic orbitals , 
 have been optimized by using the already validated and stable optimization 
 schemes discussed in ref ( 56 ) . in particular , the optimization that we have followed 
 for the singlet  and triplet states of psb3 considered 
 here in the different structures starts from an initial configuration 
 where the agp matrix is diagonal , the exponents are initialized to 
 values taken from standard dunning s basis sets ( where too 
 small and too large values are eliminated because they are not not 
 necessary due to the presence of our jastrow factor , see discussion 
 in ref ( 56 ) ) , and all 
 the jastrow parameters are set to zero , with the exception of b1 = b2 = 1 . next , 
 the optimization procedure follows the protocol : ( i ) optimization 
 of the agp , namely of the matrix elements and the contraction coefficients 
 of the basis set , with fixed exponents and jastrow parameters b1 = b2 = 1 , ( ii ) 
 optimization of the agp and relaxation of the values of the exponents 
 of the agp basis set and of the b1 and b2 parameters , ( iii ) optimization of the jastrow 
 terms , keeping the agp parameters fixed , ( iv ) optimization of the 
 overall jagp , keeping fixed the exponents in the basis set , both for 
 the agp and the jastrow , ( v ) optimization of all the parameters , including 
 the exponents of the basis set , with increasing statistical accuracy . 
 for the single diradical state (  * configuration ) of 
 psb3 in the proximity of the conical intersection , we have used a 
 slightly different procedure to avoid the possibility to be trapped 
 in a local minimum . therefore , we forced the wave function to be in 
 the correct electronic configurations by taking the triplet jagp optimized 
 wave function and obtained from that the corresponding singlet diradical 
 configuration . this wave function has been used as the starting point 
 of an optimization that started from the step ( iii ) of the previously 
 stated optimization schedule . we have verified a posteriori for every 
 nuclear structure , where we have calculated both the  and the  * singlet configurations , that the overlap 
 jagp*|jagp between the two 
 jagp wave functions ( calculated using the correlated sampling techniques ) 
 is almost zero , thus the two wave functions actually correspond to 
 different electronic states . it is important to note that qmc 
 approaches use stochastic methods 
 both to evaluate an observable and to optimize the variational parameters 
 of the wave function . the stochastic uncertainty due to the former 
 point is easy to calculate , and it has been reported in figures and 
 tables of the present work . we have carefully tested the reliability of 
 the optimization schemes used in this work , and indeed the profiles 
 reported in the following figures , although not perfectly smooth , 
 are pretty regular , both for energy and charge - transfer values . moreover , the most interesting configurations ( cis , trans , and structures close to tsct and tsdir ) have been 
 optimized with some extra effort , thus the results reported in the 
 tables are fully reliable . in this work we also report several 
 results computed at the level 
 of the fixed - node projection monte carlo scheme that has been realized 
 by performing lrdmc calculations with mesh size a = 0.3 au . although we have not performed , for computational reasons , 
 the continuous extrapolation of the lattice mesh size a 0 , we know from previous works and preliminary calculations that the bias given by the finite mesh 
 size a = 0.3 au is almost negligible in the evaluations 
 of the considered energy differences . in this section , we present the energetic 
 and electronic features 
 of the mepct , mepdir , and bla paths computed using vmc and lrdmc with 
 the jagp ansatz . a high - level treatment of electron correlation is 
 crucial for the correct description of the energy surface of the ground 
 state isomerization of psb3 . as discussed in ref ( 11 ) , dynamical electronic 
 correlation modifies the mapped casscf potential energy surface in 
 two ways : the tsct transition state is found lower in energy than 
 the tsdir , at variance with the casscf findings , and the casscf ci 
 is seen shifted to larger bla values when dynamical correlation is 
 included in the calculations . the energies along the three paths have 
 been computed using a number of electronic structure methods , namely 
 multiconfigurational approaches , multireference 
 perturbation theory , dft schemes , eom - cc , and sorci . all the most accurate methods qualitatively predict similar 
 changes in the potential energy surface with respect to casscf . in 
 particular , in the case of mrcisd+q ( the most accurate method tested 
 previously ) , tsct becomes more stable than tsdir by 4.7 kcal / mol ( compared 
 to casscf where it is less stable by 1.2 kcal / mol ) , and the ci gets 
 shifted to a bla value of 0.03  ( compared to 0.00 
  for casscf ) . before starting , we validated our computational 
 protocol by looking 
 at the electronic properties of the cis isomer of 
 psb3 such as the dipole moment  and the charge transfer of 
 the ground state s0 , defined as the partial charge on the 
 allyl moiety h2c = ch  ch= ( the net charge 
 of the system is + 1 ) . the charge transfer character at cis - psb3 is 0.313 at the casscf/6 - 31 g * level of theory and 0.355 at 
 the mrcisd+q level of theory , as derived from mulliken population 
 analyses ( see table 1 ) . because we can not define mulliken charges in our qmc framework , 
 we compute the charge transfer by finding the portion of the electronic 
 density in the region of the allyl moiety up to the plane perpendicular 
 to the c2=c3 bond and cutting it in the 
 middle . this method of obtaining the charge - transfer character is 
 tested on densities extracted from dft and wave function methods and 
 is shown to produce very similar charge transfer character as mulliken 
 charges . j1-bodysd , 
 jsd - proj , and jsd - opt are defined in the text , and the basis sets 
 for the qmc calculations are defined in section 3 . in the core column , ae stands 
 for all - electron calculation and ecp for energy - conserving pseudopotential 
 calculation . the reported numbers in the last column represent the 
 net charge on the allyl moiety . vmc and lrdmc errors on  and 
 charge - transfer are of the order of 10 . wave function optimization by dft / lda ; edft / lda = 42.6663848 h. wave function optimization by dft / lda ; edft / lda = 42.6769218 h. the single - reference nature of the 
 s0 state is highlighted 
 by the fact that charge - transfer values computed with standard dft 
 ( with pbe and b3lyp functionals ) and hf are in good agreement with 
 the mrcisd+q result . the effect of applying pseudopotentials on the 
 carbon and nitrogen atoms on the charge transfer is found to be negligible 
 when comparing all - electron ( ae ) and ecp results obtained using similar 
 basis sets , as shown in table 1 . a further 
 evidence of the reliability of our approach is given by the performance 
 of several variants of the single - determinant ( sd ) wave function ( j1-bodysd , where only the 1-body term 
 for the jastrow is used ; jsd - proj , the sd wave function is projected 
 out from the full agp ; jsd - opt , the sd wave function is optimized 
 after projection ) in the vmc framework ( e.g. , taking into account 
 only the first term of the agp expansion in eq 14 ) that , using ecp , are in full agreement with the more accurate vmc / jagp 
 ( 0.358 , the best variational result ) and the mrcisd+q . lrdmc / jagp 
 only slightly corrects ( 0.352 ) the vmc / jagp result for the charge 
 transfer . as first , 
 we consider the energy difference between the cis and trans isomers of psb3 . vmc / agp and lrdmc / agp 
 values ( table 2 ) , 2.9(2 ) and 3.0(2 ) 
 kcal / mol respectively , and the vmc / jsd value ( 2.9(2 ) kcal / mol ) 
 are fully consistent with the reference mrcisd+q ( 3.1 kcal / mol ) , xmcqdpt2 ( 2.8 kcal / mol ) , and 
 eom - ccsd ( 3.0 kcal / mol ) . the 
 negative value indicates that trans - psb3 is more 
 stable than cis - psb3 because throughout this work 
 the cis - psb3 energy is taken as the reference . t between the first singlet and the first triplet electronic states 
 of the cis isomer is also reported . the reported 
 qmc results are compared with mrcisd+q , xmcqdpt2 , and eom - ccsd calculations . at vmc / jagp level , the tsct 
 ( 45.5(2 ) kcal / mol ) is lower in energy 
 than the tsdir ( 51.7 kcal / mol ) , making the ct path energetically favored , 
 similarly to what was reported by the aforementioned correlated approaches . lrdmc and vmc findings are equal within the stochastic error to 
 the vmc / jagp values , evidence that the trial wave function t is fully optimized . the singlet  triplet 
 gap for the cis - psb3 is also reported , with a difference 
 of 2.3(3 ) kcal / mol between vmc and lrdmc using the complete jagp . the vmc / jagp energy profile of the mepct path ( figure 3a ) is characterized by a shallow plateau around 
 the transition state structure , at variance with the shape of the 
 mepdir path ( figure 3c ) ; moreover , expensive 
 lrdmc calculations do not alter the picture . parts b and d of figure 3 show the ratio between lumo and 
 homo , according to the agp expansion given in eq 14 ; as already discussed by some of us in the case 
 of the application of the agp ansatz on the diradical twisted ethylene 
 c2h4 , very small 
 values of this ratio correspond to a single - reference wave function , 
 with the lowest molecular orbitals doubly occupied ( the ratio is exactly 
 zero in the limit of a pure single slater determinant ) , whereas large 
 values of the ratio indicate two ( near)-equivalent configurations . 
 following the analysis in ref ( 32 ) , the absolute value of lumo/homo approaching to unity means that two configurations are 
 contributing with the same weight to the electronic structure . homo and lumo correspond to 1 and 2 in the simple ( 2,2 ) model in eq 18 . the wave function along the mepct path ( figure 3b ) is dominated by a single configuration , with 
 the lumo/homo oscillating around 
 the value for the cis isomer ( 0.0635 ) , as 
 expected by the previous investigations on the mepct path of psb3 . the same behavior has been found along the 
 mepdir path ( figure 3b ) , with no appreciable 
 contribution given by higher - energy configurations ; this result unequivocally 
 shows that the introduction by qmc of balanced dynamical correlation 
 strongly alters the casscf description of the electronic structure 
 of psb3 along the mepdir path , similarly to what found in mrcisd+q 
 calculations . left column : ( a ) thermal isomerization energy profile 
 ( with respect 
 to the cis psb3 ) and ( b ) ratio between lumo and homo ( see eq 14 ) along the mepct reaction coordinate . right column : ( c ) thermal 
 isomerization energy profile and ( d ) ratio between lumo and homo along the mepdir reaction coordinate . sd stands for single determinant , e.g. , a single configuration ( with 
 ratio lumo/homo exactly equal 
 to zero ) . error bars are within the symbols . for both paths casscf 
 and mrcisd+q energy profiles are taken from refs ( 11,12 ) . the agp wave function can be formally 
 expanded into a linear combination 
 of slater determinants ( eq 14 ) , so the single - electron 
 molecular orbitals are obtained by the diagonalization of the geminal 
 coupling matrix ( eq 10 ) and are defined within 
 a correlated framework . the agp spans the seniority number  
 = 0 sector in hilbert space , with double occupation for each orbital . the terms  closed - shell  and  open - shell  
 are widely used to indicate systems without and with unpaired electrons , 
 respectively , implying that the molecular orbitals come from single - reference 
 approaches , like hartree  fock or dft . in the case of highly 
 twisted configurations of psb3 , for instance , the  state , involving charge transfer with respect to the equilibrium 
 ground state , has closed - shell character , while the diradical  * 
 state has open - shell character : for  * static correlation 
 plays an important role , and a multiconfigurational approach must 
 be used . such definitions strictly depend on the choice of the reference 
 for the molecular orbitals , as explained in section 2.3 . even though the agp wave function is formally characterized 
 by only doubly occupied orbitals , its application is not limited so 
 far to the study of closed - shell systems because the molecular orbitals 
 involved in the agp expansion are the results of a variational optimization 
 and they can be qualitatively different from the hartree data 
 show that the mepdir path does not have a diradical character anymore ; 
 the vmc / jagp description of mepct and mepdir paths produces a ground 
 state of closed - shell  character , similarly to 
 the mrcisd+q . the results above 
 are again reflected in the charge transfer profiles 
 along both mepct and mepdir paths shown in figure 4 . indeed , it is clear that in both mepct and mepdir , the wave 
 function gains charge transfer character , with tsct and tsdir both 
 having ca . the picture 
 presented at the vmc / jagp and lrdmc / jagp levels is again at variance 
 with what is found at the casscf level of theory . with casscf , mepdir 
 passes through a transition state , tsdir , which has diradical character 
 and has almost no charge transfer at all , with all the charge localized 
 on the nitrogen - containing moiety . this difference is due to the change 
 in ci position on the energy surface after introducing a balanced 
 description of electronic correlation in the calculations . at the 
 casscf level of theory , the ci is peaked and lies in between the tsct 
 and tsdir transition states , causing them to have wave functions with 
 different character . at correlated levels of theory , the ci is shifted 
 to a larger bla value than that corresponding to tsdir . as a result , 
 both tsct and tsdir lie on the same side of the ci ( which is now intermediate / sloped ) 
 and therefore both have the same wave function character ( charge transfer ) . this change in local topology causes only one of the transition states 
 to remain as a saddle point , tsct , while tsdir is no longer a transition 
 state on the s0 potential energy surface . the charge transfer 
 character in both mepct and mepdir decreases as psb3 moves toward 
 the cis or trans isomers of psb3 , 
 converging to the values corresponding to the two minima . casscf and mrcisd 
 charge - transfer profiles are taken from refs ( 11,12 ) . furthermore , the mepct maximum 
 ( i.e. , the tsct ) region is flatter 
 for the two qmc levels than for mrcisd+q level ; this may be due to 
 a better treatment of the electronic dynamical correlation in this 
 region . in the casscf(6,6)/6 - 31 g * 
 landscape , the ci is located between the tsct and tsdir . consistently 
 with the other approaches , psb3 assumes a closed - shell , charge transfer 
 character (  ) at smaller bla values than the ci , 
 while its wave function becomes covalent and diradical (  * ) 
 at larger bla values ( upper panel of figure 5 ) . as anticipated above , the crossing between the two states does 
 not produce a peaked ci , like in the casscf case , but instead a sloped 
 and intermediate ci . even though we are not able to identify the exact 
 position of the ci , vmc / jagp calculations clearly show that ci moves 
 toward large values of bla ( 0.075  ) , beyond the tsdir . the present result differs from the collection of data obtained by 
 other correlated approaches , locating the ci around 0.030.04 
  , with the exception 
 of the qd - nevpt2/cas(6,6 ) analysis ( extrapolated value at 0.05  ) . when using the lrdmc / jagp approach , the ci position 
 ( 0.06  ) comes closer to mrcisd+q data . we observe that 
 the charge transfer (  ) curves obtained using vmc / jagp 
 and lrdmc / jagp are actually more stable , with respect to cis - psb3 energy , than that obtained with mrcisd+q , consistently with 
 the lower energy tsct of vmc / jagp and lrdmc / jagp . meanwhile , the diradical 
 (  * ) curve from vmc / jagp or lrdmc / jagp are less stable 
 that those from mrcisd+q . this is what causes the ci from vmc / jagp 
 and lrdmc / jagp to shift to higher bla than in mrcisd+q . moreover , 
 it is important to point out that relaxing the structures and the 
 minimum energy paths of psb3 at the qmc correlated level may produce 
 slightly different topologies for the bla and mep scans close to the 
 ci . energy profile along the bla path ( with respect to the cis psb3 ) and the related lumo/homo ratio for the  and  * 
 configurations , calculated by vmc and lrdmc methods on the variationally 
 optimized jagp wave functions . the mrcisd+q , mrcisd , and casscf profiles are shown , for 
 comparison , and the conical intersections ( ci ) obtained with the different 
 approaches are marked in the plot . in the lower panel of figure 5 , the 
 vmc / jagp 
 points corresponding to the  * state are characterized 
 by a high ratio ( in absolute value ) between homo and lumo , close to 1 . as already mentioned , this 
 is due to the multiconfigurational nature of the diradical state , 
 where the frontier orbitals are near - degenerate and singly occupied . 
 from the agp analysis , near - degeneracy for s0 is therefore 
 found in the large - bla portion of the path . this reinforces the results 
 of the mepct and mepdir paths above because it is clear here that 
 tsct and tsdir both lie on the same side of the ci after introduction 
 of the dynamical electronic correlation . also , one can see here that 
 while tsct remains a saddle point on the s0 potential energy 
 surface , tsdir actually becomes a minimum on the s1 surface 
 and is no longer a transition state as in casscf . the  state , also extending in the diradical portion of space previously 
 defined by the casscf study , is clearly single - reference , and it dominates 
 the wave function of psb3 up to a bla of 0.05  . the reliability 
 of the present results is further confirmed looking 
 at the triplet energy profile along the bla path ( blue triangles in 
 the upper panel of figure 5 , filled and open 
 symbols for vmc and lrdmc , respectively ) . because the comparison of 
 the  and  * energies along the bla 
 scan is a fundamental step in order to give an accurate representation 
 of the ground state potential energy surface of psb3 surrounding the 
 ci , the triplet energy profile can be considered a lower bound for 
 the  * energy . for diradicals , triplet should be the 
 ground state spin multiplicity , according to the molecular version 
 of hund s rule : this is the case , for instance , 
 of the orthogonally twisted ethylene molecule where , due to the homolytic cleavage of the double bond , the wave 
 function is dominated by two configurations with the same weight and 
 the two involved p atomic orbitals have zero overlap . the structures of the bla path with diradical character are characterized 
 by a torsion of about 90 deg around the formal central double bond 
 of psb3 , similar to the prototypical example given by c2h4 system : the central double bond is broken and the two p orbitals are mutually ( almost ) perpendicular . this manifest 
 similarity between the twisted diradical psb3 and the prototypical 
 c2h4 system yields to reasonably expect that 
 the ground state of the twisted diradical psb3 is also a triplet , 
 as we observe a posteriori with vmc / jagp and lrdmc / jagp calculations . [ however , it is not true in general for all diradicals that the triplet 
 state has a a lower energy that the singlet state , for example , disjointed 
 and non - kekule molecules , like the tetramethyleneethane , have a very small singlet  triplet gap with the singlet lower 
 in energy in some specific geometries , while the ground state of oligacenes 
 larger than hexacene is also a singlet . ] as one can see from the upper panel of figure 5 , the triplet energy is higher than the  singlet energy for bla values smaller than 0.055  for vmc 
 and 0.05  for lrdmc , and consequently the same certainly occurs 
 for the  * state . a similar gap of 1012 
 kcal / mol is also found at casscf(6,6)/6 - 31 g * level in the diradical 
 portion of the bla path . these simple arguments , combined with 
 the fact that the convergence 
 of the wave function optimization for a triplet state is easier to 
 achieve in the vmc / agp framework , allow us to be extremely confident 
 with the robustness of the present results for the location of ci 
 along the bla path . the ground state wave function is dominated 
 by the  state before the ci ( i.e. , at lower bla 
 values ) , while it 
 assumes  * character after the crossing ; it is interesting 
 to understand how the charge distribution changes in the two configurations . figure 6 reports , as an example , the charge 
 density difference between the  and  * 
 states for vmc / jagp and lrdmc / jagp ci points along the bla path , more 
 precisely , for the closest structures to ci found at vmc and lrdmc 
 level . the shape of the isosurfaces obviously resembles the p atomic orbitals involved in the electronic structure 
 of the twisted psb3 . the most evident changes in the charged distribution 
 are observed for the two central carbon atoms , where the double bond 
 has been broken because of the torsion : the yellow part of the plot 
 ( online color version ) corresponds to an accumulation of electronic 
 charge in the  state with respect to the  * , 
 whereas the green isosurfaces indicate the opposite situation . graphical representation 
 of the charge density difference between 
 the  and the  * configurations for 
 vmc / jagp and lrmdc / jagp ci points : the yellow ( green ) isosurface indicates 
 an excess ( reduction ) of 0.01 in terms of electronic charge in the 
  state with respect to *. such analysis is consistent with the charge transfer 
 nature of 
  configuration and with the covalent and diradical 
 character of *. we know from figure 7 that the charge transfer of  along the 
 path oscillates between 0.63 and 0.72 , and that the charge transfer 
 of  * is much smaller ( 0.10 ) . a large ( small ) 
 value of charge transfer means an excess ( reduction ) of electronic 
 charge on the protonated imine heteroallyl moiety , according to the 
 plot reported in figure 7 for vmc and lrdmc 
 calculations , with the charge transfer of the triplet coinciding with 
 the  * values . charge transfer along the bla path for the  , 
  * , and triplet configurations , obtained by vmc and lrdmc 
 approaches ( stochastic errors are smaller than point size ) . vmc and lrdmc methods have 
 been used to compute electronic and 
 energetic properties of psb3 , using the jagp ansatz . vmc / jagp 
 and lrdmc / jagp calculations on three different casscf 
 paths reveal the fundamental role played by a balanced description 
 of the dynamical correlation for the correct representation of the 
 ground state energy surface in the proximity of the conical intersection . the vmc and lrdmc calculations significantly alter the casscf landscape , 
 inverting the relative stability of the mepct and mepdir paths , similarly 
 to what obtained by other correlated approaches . however , the region surrounding the tsct appears 
 more flat when computed at the qmc levels . on both mepct and 
 mepdir paths , the psb3 wave function assumes 
 a charge transfer and single - reference character : the mepdir path 
 is therefore not diradical anymore when analyzed at correlated level . this finding is immediately confirmed by the fact the ci is pushed 
 toward values of bla larger than that corresponding to the tsdir ( 0.075 
  with vmc / jagp and 0.06  with lrdmc / jagp , while 
 the reference calculations give a ci located at around 0.030.04 
  ) . furthermore , in the present study , the multiconfigurational 
 nature 
 of agp has been explained in detail , following the work reported in 
 ref ( 32 ) , for the description 
 of the diradical  * configuration of psb3 . even though 
 the agp is defined in the seniority number  = 0 subsector of 
 the hilbert space ( e.g. , the set of molecular orbitals is doubly occupied ) , 
 a unitary transformation of the agp - optimized molecular orbitals allows 
 one to always map the  * state of highly twisted psb3 
 geometries , with two unpaired electrons , into the agp  = 0 
 subsector . the role of the dynamical electronic correlation 
 has been found 
 to be essential in order to get a reliable description of the ground 
 state of psb3 around the ci . therefore , mechanistic or dynamics studies 
 using methods which do not incorporate these effects need to be performed 
 and interpreted critically . the recent improvements in forces calculations 
 using qmc approaches suggest that there 
 will be soon the possibility to compute mepct and mepdir minimum energy 
 paths based on the vmc / jagp method , yielding to a more consistent 
 comparison with the energies and geometries obtained from casscf in 
 order to further clarify the main features of psb3 .
the 
 penta-2,4-dieniminium cation ( psb3 ) displays similar ground 
 state and first excited state potential energy features as those of 
 the retinal protonated schiff base ( rpsb ) chromophore in rhodopsin . 
 recently , psb3 has been used to benchmark several electronic structure 
 methods , including highly correlated multireference wave function 
 approaches , highlighting the necessity to accurately describe the 
 electronic correlation in order to obtain reliable properties even 
 along the ground state ( thermal ) isomerization paths . in this work , 
 we apply two quantum monte carlo approaches , the variational monte 
 carlo and the lattice regularized diffusion monte carlo , to study 
 the energetics and electronic properties of psb3 along representative 
 minimum energy paths and scans related to its thermal cis  trans isomerization . 
 quantum monte carlo 
 is used in combination with the jastrow antisymmetrized geminal power 
 ansatz , which guarantees an accurate and balanced description of the 
 static electronic correlation thanks to the multiconfigurational nature 
 of the antisymmetrized geminal power term , and of the dynamical correlation , 
 due to the presence of the jastrow factor explicitly depending on 
 electron  electron distances . along the two ground state isomerization 
 minimum energy paths of psb3 , casscf calculations yield wave functions 
 having either charge transfer or diradical character in proximity 
 of the two transition state configurations . here , we observe that 
 at the quantum monte carlo level of theory , only the transition state 
 with charge transfer character can be located . the conical intersection , 
 which becomes highly sloped , is observed only if the path connecting 
 the two original casscf transition states is extended beyond the diradical 
 one , namely by increasing the bond - length - alternation ( bla ) . 
 these 
 findings are in good agreement with the results obtained by mrcisd+q 
 calculations , and they demonstrate the importance of having an accurate 
 description of the static and dynamical correlation when studying 
 isomerization and transition states of conjugated systems .