C Hentzen

N Turmel

C Chesnel

G Miget

F Le Breton

A Charlanes

E Tan

G Mrs

Amarenco

What Criteria Affect a Patient's Choice of Catheter for Self-Catheterization?

Keywords: Intermittent Urethral Catheterization, Patient Preference, Patient Education as Topic, Teaching, Nurses, Urinary Retention, Urinary Bladder, Neurogenic Registered on clinicaltrials.gov: NCT04033913

Aims: The aim was to determine which criteria are most influential in i in the atient's choice of catheter during clean intermittent self-catheterization (CISC) education.

Methods: a questionnaire exploring five traits (catheter design, catheter length, comfort of use n rse's ex lanations and how easy it was to carry and dispose of the catheter) was administered to all patients who succeeded in performing CISC. The patients had to report the criterion that most influenced their choice, and rate the importance of each criterion from 1-4 on a Likert scale (strongly disagree, somewhat disagree, somewhat agree, strongly agree). The study assessed the impact that age, sex, etiology, dependency, motor or sensory upper limbs deficiency and position required to perform CISC may have had on the importance of the different criteria.

Results: Seventy-three patients were included (mean age 50.9 ± 14.3 years). The most important criterion was the nurse's explanations (44%), followed by comfort of handling (32%), the discreet aspect of carrying the catheter (15%), catheter length (7%). The esthetic aspect was not important for 34 % of patients, whereas all the other criteria were important for at least 82% of patients. Comfort of handling was more often a priority for those with motor or sensory upper limbs disorder (p<0.01). The other characteristics of the patients did not impact the way they prioritized the criteria.

Conclusion: These results highlight the importance of extensive training for nurses involved in CISC education, as their explanations have the greatest impact on the patient's choice of catheter.

Introduction

Clean intermittent self-catheterization (CISC) was introduced by Lapides et al in 1972 [START_REF] Lapides | Clean, intermittent self-catheterization in the treatment of urinary tract disease[END_REF] .

Nowadays, it is the gold standard for the treatment of urinary retention in neurogenic bladders and is being proposed more and more often to patients with non-neurogenic lower urinary tract symptoms (LUTS). Several studies have demonstrated an improvement in the quality of life of patients using CISC [START_REF] Johansen | A Novel Product for Intermittent Catheterisation: Its Impact on Compliance with Daily Life-International Multicentre Study[END_REF][START_REF] Bakke | Social functioning and general well-being in patients treated with clean intermittent catheterization[END_REF] . The availability of hydrophilic and self-lubricating catheters has reduced traumatic and infectious complications and compact catheters have contributed to improved quality of life, allowing smoother integration of CISC into the atient's social an work life [START_REF] Chartier-Kastler | A prospective, randomized, crossover, multicenter study comparing quality of life using compact versus standard catheters for intermittent self-catheterization[END_REF] . Some studies have been conducted to explore factors limiting CISC, whether difficulties learning or in continued use. These studies found that obesity, being female, or having motor or cognitive disorders have a negative impact on the ability to learn CISC [START_REF] Vahter | Clean intermittent selfcatheterization in persons with multiple sclerosis: The influence of cognitive dysfunction[END_REF][START_REF] Castillo | Urologic self-management through intermittent self-catheterization among individuals with spina bifida: A journey to selfefficacy and autonomy[END_REF][START_REF] Hentzen | Intermittent Self-catheterization in Older Adults: Predictors of Success for Technique Learning[END_REF] , and older age, important stress urinary incontinence, and dependency for CISC installation were predictive for a poorer adherence to the method [START_REF] Afsar | Compliance with clean intermittent catheterization in spinal cord injury patients: A long-term follow-up study[END_REF]9 .

Aseptic or clean intermittent catheterizations are taught to patients, with no evidence of the superiority of one over the other in terms of occurrence of complications. Guidelines on technique differ according to scientific societies. Teaching self-catheterization is now well established, and various nursing journals describe the different steps of the CISC educational program [START_REF] Mangnall | Managing and teaching intermittent catheterisation[END_REF] . However, no studies have shown the benefit of these programs on successful learning or adherence [START_REF] Breton | Therapeutic education and intermittent self-catheterization: Recommendations for an educational program and a literature review[END_REF] . Many psychological barriers can limit the practice of CISC [START_REF] Guinet-Lacoste | Intermittent catheterization acceptance test (I-CAT): A tool to evaluate the global acceptance to practice clean intermittent self-catheterization[END_REF] but the main challenge is to obtain the patient's acceptance and long-term adherence to CISC, as urinary disorders are mostly chronic, and sometimes involve a risk of damage to the upper urinary tract specially in neurogenic patients like those with spinal cord injuries and multiple sclerosis.

Currently there are many catheter models available. The choice may depend on different factors: grip disorder, ease of use, design and packaging, discomfort or pain when testing a specific material, specific anatomical conditions (male Tiemann or olive tip), longer catheter length in case of imperfect emptying, explanations given on how to use the catheter, etc.

While the doctor and nurse involved in the patient's therapeutic education guide the choice of catheter according to medical criteria and the patient's physical capacities (despite the lack of objective and randomized studies for many criteria), the patient remains an active participant in the choice of model. Indeed, being involved in this process could encourage adherence to the treatment.

The primary aim of this study was to evaluate which criteria are most important to the patient when choosing a catheter. The secondary aim was to assess whether the individual characteristics of the patient changed the importance he or she attached to each criterion.

Methods

Study design

This study was approved by the local ethics review board, registration number 2018-A01644-51, and all participants provided written informed consent before inclusion in this observational study. This study was registered on clinicaltrials.gov: NCT04033913.

Participants were recruited in the Neuro-urology department, the day they came to learn to perform CISC (educational program).

Participants

Inclusion criteria were: adult patients who have successfully perform self-catheterization during the educational program in the Neuro-urology department. Exclusion criteria were only refusal to participate. Participants were recruited between November 2018 and August 2019. Patients were included and completed the questionnaire on the day of the CISC training, so they had not yet tried CISC in their daily life.

Procedure

The questionnaire was developed based on meetings with experts and reviewing literature about what factors may contribute to patient satisfaction with their self-catheterization device. The panel of experts was composed of 7 doctors specialized in neuro-urology, 3

nurses with significant experience in CISC therapeutic education, one occupational therapist and one neuropsychologist involved in therapeutic education. Based on literature review, 7 criteria were singled out: " esi n of the catheter" " acka in a earance" "len th of the catheter" "comfort han lin the catheter" "comfort intro cin the catheter" "n rse's ex lanations" "easy to wear an is ose of the catheter". Base on clinical ex erience the criterion "comfort when intro cin the catheter" was eliminate eca se the majority of patients try only one type of catheter after handling several, and therefore cannot compare comfort when introducing it into the urethra between several models. The experts also decided to combine the design of the catheter and the appearance of the packaging into a sin le criterion "catheter an acka in esi n"; the two criteria seem too similar to be distinguished.

Five factors were therefore finally selected: "catheter and packaging design", "catheter length", "comfort using the catheter", "n rse's ex lanations", "easy to carry and dispose of the catheter".

This led to a short questionnaire with 5 questions:

-Position in which CISC was performed (toilet, wheelchair, bed) The importance of each criterion was rated from 1-4 on a Likert scale (strongly disagree, somewhat disagree, somewhat agree, strongly agree), and each patient had to report the most important criterion in the choice of the catheter. In this case, there was no need for a neutral mid-point on the scale because the question was not about satisfaction with a criterion, but about its importance in the atient's choice. Thus, a neutral mid-point was irrelevant.

-
Nurses gave the questionnaire at the end of the training. Demographic data was collected (age, sex, etiology of urinary disorders) and some functional scores were taken during therapeutic education to assess dependency (Functional Independence Measure (FIM) score [START_REF] Ottenbacher | The reliability of the functional independence measure: A quantitative review[END_REF] ), sensory-motor functions in the upper limbs (Motor hand disorders assessed by a Nine Hold Peg Test 14 over 18, sensory hand disorders assessed by Weber compass and clinical examination), and CISC skills (Pencil and Paper Test [START_REF] Amarenco | Pencil and Paper Test: A New Tool to Predict the Ability of Neurological Patients to Practice Clean Intermittent Self-Catheterization[END_REF] , maximum score of 15)

Statistical analysis

Statistical analyses were performed with the R software for Windows (Rx64 3.2.3, R Foundation for Statistical Computing, Vienna, Austria). Descriptive data were presented as means with standard deviation for continuous data and as medians with range for ordinal data and data not normally distributed. For secondary analyses, the res onses "stron ly isa ree" an "somewhat isa ree" were oole into " isa ree" an the responses "stron ly a ree" an "somewhat a ree" into "a ree" to i entify associations with atients' characteristics. The association between patient characteristics and the choice of priority criterion was studied only for the 2 criteria most commonly identified as the most important.

The impact of age, sex, etiology, score on the Pencil and Paper Test, dependency, motor or sensory deficit of the upper limbs, and the position in which CISC was performed on the importance of the different criteria was assesse sin Welch's t-test and the Fisher Test.

Significance was taken as p < 0.05.

Results

Patients characteristics

Seventy-three patients were included, with a mean age of 50.9 ± 14.3 years. Patient characteristics are provided in Table 1. The majority of patients (89%) learned to perform CISC on the toilet (sitting or standing), 5.5% learned in a wheelchair and 5.5% in a bed. The majority of patients (97%) had seen at least two models of catheter, on average 2.8 ± 1.2 were shown. Forty-six (63%) patients tried at least two different models of catheter.

Forty-five patients (62%) chose the first model they tried, 18 (25%) the second one, 5 (7%) two models to adapt to their activities.

Fifty-nine patients fully completed the questionnaire and 14 patients did not choose a most important criterion. All the other questions were completed.

Most important criterion in catheter choice

The most important criterion for the final choice of catheter was the nurse's explanations (44%), followed by the comfort of handling the catheter (32%). The discreet aspect of transport and disposal was a priority for 15% of patients, the length of the catheter for 7%, and the esthetic aspect only for 1 patient (2%). Results are summarized in Figure 1. Patients who did not answer this question were not different from those who completed the entire questionnaire, except at a lower mean age (44.2 ± 12.1 vs 52.5 ± 14.4, p=0.04). The comparison of the two groups is reported in Table 4 in supporting information.

Importance of each criterion

The way the catheter looked (design) was not important to 34 % of patients, whereas all the other criteria were important for at least 82% of them. The n rse's ex lanations were the only criterion of unanimous importance. All the answers are reported in Figure 2.

Impact of patient characteristics on the criterion they prioritized

The comfort of handling the catheter was more often a priority for patients with sensory and/or motor deficits in the upper limbs (p<0.01). Age, sex, etiology of LUTS, Pencil and Paper Test fewer than 15, FIM score and position in which CISC was performed did not impact the choice of atients' prime criterion. All results are reported in Table 2.

Impact of patient characteristics on the importance of each criterion

Women were more likely to attach importance to catheter and packaging design than men (p=0.03). In contrast, patients who catheterized themselves in bed or in a wheelchair gave less importance to this criterion (p=0.02). Catheter length was important for 88% of women and only 60% of men (p = 0.03).

Concernin the criterion "easy to carry an is ose of the catheter" the n m er of ifferent catheters seen were higher for patients who considered this criterion to be important (2.3 ± 0.9 vs. 2.9 ± 1.2; p=0.04).

Other patient characteristics did not influence the importance given to each criterion. All the results are reported in Table 3.

Discussion

Our study shows that in general all the criteria studied are important in the patient's choice of catheter. Discrete transportation and disposal are more important than the design of the catheter. The most important criterion is the n rse's ex lanations followe y ease of use and handling, especially for patients with sensory and/or motor deficits of the upper limbs.

These results are not surprising since the different criteria mentioned are those explored in the questionnaire InCaSaQ on catheter satisfaction, defined on the basis of an open-ended interview with patients who performed CISC [START_REF] Guinet-Lacoste | Validation of the InCaSaQ, a new tool for the evaluation of patient satisfaction with clean intermittent self-catheterization[END_REF] . rse's ex lanations were often cited as the most important criterion. There is potentially a bias in these responses; the questionnaire was given by the nurse at the end of the training, and the patient might want to please the nurse/doctor with his or her answer. However, there are several other explanations for this choice. First of all, the nurse is a healthcare professional, specifically trained to provide therapeutic CISC education. He/she has specific knowledge that allows him/her to sum up the pros and cons of each type of catheter. Secondly, patients are unfamiliar with CISC so they rely on the person who has the expertise. Logan et al [START_REF] Lo An | Patients' ex eriences of learnin clean intermittent self-catheterization: A qualitative study[END_REF] reported with patient interviews how learning CISC can be stressful and embarrassing. Nurses' attitudes, including a friendly relationship and sufficient explanations are one of the main elements that promote a positive experience in CISC therapeutic education. In our study, of the 30 patients who left a free-form comment, 11 thanked or highlighted the nurse's professionalism.

The need for easy use has been described by Denys et al [START_REF] Denys | Intermittent self-catheterization habits and opinion on aseptic VaPro catheter in French neurogenic bladder population[END_REF] as a factor that can lead patients using CISC to change their catheter type. It seems natural that when learning a new technique that may be quite stressful, an easy-to-handle catheter should be an important criterion. The importance of discretion when carrying and disposing of the catheter was evoked by Jaquet et al [START_REF] Jaquet | Coping with clean intermittent catherizationâ" x eriences from a atient ers ecti e[END_REF] as paramount for patients who had been performing CISC for at least 6 months. Patients reported the importance of being able to hide the catheter in public places and they did not mention CISC at work, reflecting their desire for the discreet social integration of their CISC device.

Women attached more importance than men to esthetics and more surprisingly to the length of the probe. One possible explanation is that male catheters are all relatively long, because of the length of the male urethra, while women may be "pleasantly surprised" when they discover that female catheters are relatively short, far from the image of indwelling catheters that they might be familiar with. The impact of enlarged prostate on catheter choice was not studied because among the men included, 4 had prostate hyperplasia diagnosed with ultrasonography, but this obstruction was not the cause of urinary disorders for any of them (detrusor underactivity or detrusor sphincter dyssynergia).

This study is one of the only ones interested in the patient's perception of the different types of catheters during therapeutic education with a large sample. The interest is multiple; at present, the choice of catheter models is so vast that it is inevitable that the patient's choice be taken into account. The patient's participation in this choice makes him/her an "actor" in his/her treatment, and could promote adherence to CISC. In cases of multiple sclerosis, the American Association of Neurology recommends at a grade A level incorporating patient reference in the choice of " isease mo ifyin thera y" s estin that taking into account patient preferences may improve acceptance of and adherence to "disease modifying thera y" [START_REF] Day | Practice guideline recommendations summary: Disease-modifying therapies for adults with multiple sclerosis: Report of the Guideline Development, Dissemination, and Implementation Subcommittee of the American Academy of Neurology[END_REF] . he conce t of "share ecision-makin " has been expanding for several years in the treatment of chronic diseases. It is a process in which the hysician elicits the atient's values and preferences about their care, has an evidence-based discussion of treatment options, and allows for a common treatment decision to be reached [START_REF] Colligan | Shared decision-making in multiple sclerosis[END_REF] . This approach showed several benefits in patient satisfaction and knowledge. The positive influence on adherence is less systematic and has been demonstrated in hypertension [START_REF] Schoenthaler | Medication Adherence Improvement Similar for Shared Decision-Making Preference or Longer Patient-Provider Relationship[END_REF] and asthma [START_REF] Kew | Shared decision-making for people with asthma[END_REF] but not yet in neurological diseases such as multiple sclerosis [START_REF] Colligan | Shared decision-making in multiple sclerosis[END_REF][START_REF] Ben-Zacharia | Impact of Shared Decision Making on Disease-Modifying Drug Adherence in Multiple Sclerosis[END_REF] nor in neuro-urology. Better knowledge of which criteria are important to the patient can improve therapeutic education, but also provide a guide for the criteria to be met in the creation of new catheter models.

This study has several limitations. First, the questionnaire used is not validated but was the result of an expert consensus. Secondly "n rse's ex lanations" are a variable factor, depending on the habits, experience, preferences and perceptions of each healthcare professional. The presentation of the catheters will generally be adapted to each patient, depending on the transference and counter-transference of the situation. However, all the nurses participating in this session were very experienced with over 4 years of clinical practice in teaching CISC. Another limitation is the heterogeneity of the population, with a majority of women (79%) and neurologic patients (89%). But this probably reflects the Table 3: Impact of patient characteristics on the importance of each criterion he criterion "n rse's ex lanations" is not mentione since all atients consi ere it important LUTS: Lower urinary tract symptoms; sd: standard deviation; FIM: Functional Independence Measure; N. central: Neurological central; N. peripheral: Neurological peripheral; Imp: Im ortant criterion for the choice corres on in to oole res onses "stron ly a ree" an "somewhat a ree"; ot im : Criterion not important for the choice, corresponding to pooled res onses "stron ly isa ree" an "somewhat isa ree"

  Number of types of catheter showed during therapeutic education -Importance of the different factors in the choice of catheter -Number of types of catheter tried during the therapeutic education -Choice of catheter (the first one tried, the second one, two different types, other)

  population affected by CISC whose main indication is neurological urinary retention, with a large prevalence of multiple sclerosis. The answers obtained are subjective and can probably also be influenced by factors other than those tested (patient's environment, psychological state, duration of follow-up in the department). Different perspectives exist. The importance given to the nurse's message supports the need for learning CISC during a standardized therapeutic education program. Oh et al 25 reported greater patient satisfaction when CISC was learned during a specific and standardized program, with better understanding and assimilation of the provided information. The explanations should allow the patient to choose the catheter best suited to his/her physical abilities and daily activities. It would be interesting to explore the long-term satisfaction (weeks, months, years) of patients with the catheter they chose based on the most important criterion in their choice. Even more than patient training and information, this study highlights the need for specific and detailed training of the nurses involved in therapeutic education programs. It is essential that they have a good understanding of the different characteristics of the catheters, and can present their advantages and disadvantages, without being influenced by the marketing arguments put forward by sales representatives.5. ConclusionA atient's choice of catheter type is strongly influenced by the explanations given by the nurse during therapeutic education. Comfort in handling is the second most important criterion. More important than the esthetics of the packaging and the catheter, is its ability to be carried discreetly and integrated into the patient's social life. These results highlight the importance of extensive training for nurses involved in CISC therapeutic education programs.

Figure 1 :Figure 2 :

 12 Figure 1: Most important criterion in the choice of catheter designated by each patient

Table 1 : Characteristics of study participants

 1

	LUTS: Lower urinary tract symptoms
	Sd: standard deviation
	Participants
	Age, years ± sd	50.9 ± 14.3
	Gender
	Female	58 (79%)
	Male	15 (21%)
	Neurologic etiology of LUTS
	Central	40 (58%)
	Peripheral	16 (23%)
	Etiology
	Multiple sclerosis	27 (37%)
	Spinal cord injury	6 (8%)
	Cauda equina syndrome or neuropathy	14 (19%)
	Neurological -others	7 (10%)
	Non neurological	11 (15%)
	Not determined	8 (11%)
	Number of catheter models shown, mean ± sd	2.8 ± 1.2
	Pencil and Paper Test (median, range)	15 [11-15]
	Sensory and/or motor hand disorders	35 (48%)
	Functional Independence Measure score (median, range)

Table 2 : Impact of patient characteristics on their priority criterion

 2 Only the results for the two designated priority criteria are represented LUTS: Lower urinary tract symptoms; sd: standard deviation; FIM: Functional Independence Measure; Priority: criterion identified as the most important for the choice of the catheter of the 5 proposed; Not: Another criterion identified as the most important

		Comfort of using the catheter		Nurse's explanations
		Priority n=19	Not n=40	p	Priority n=26	Not n=33	p
	Gender
	Female	16 (84%)	32 (80%)	1	20 (77%)	28 (85%)	0.52
	Male	3 (16%)	8 (20%)		6 (23%)	5 (15%)
	Etiology of LUTS
	Non-neurological	2 (11%)	10 (27%)	0.40	8 (31%)	4 (13%)	0.07
	N. central	12 (67%)	20 (54%)		16 (61%)	17 (57%)
	N. peripherical	4 (22%)	7 (19%)		2 (8%)	9 (30%)
	Sensory or motor hand disorder
	Yes	13 (72%)	12 (33%)	< 0.01	10 (42%)	15 (50%)	0.60
	No	5 (28%)	24 (67%)		14 (58%)	15 (50%)
	Pencil and Paper Test
	15	14 (82%)	28 (74%)	0.74	16 (67%)	26 (84%)	0.21
	< 15	3 (18%)	10 (26%)		8 (33%)	5 (16%)
	Position
	On the toilets	17 (90%)	37 (92%)	0.80	23 (88%)	31 (94%)	0.79
	Wheelchair	1 (5%)	1 (3%)		1 (4%)	1 (3%)
	Bed	1 (5%)	2 (5%)		2 (8%)	1 (3%)
	Age	55.3±14.5	51.2±14.3	0.32	54.5±12.7	50.9±15.6	0.34
	(mean ± sd)
	Number of catheters seen	2.9±1.2	2.8±1.1	0.74	2.7±1.2	2.9±1.1	0.48
	(mean ± sd)
	FIM score	103.0±10.8	104.9±13.0	0.61	103.6±13.8	105.0±10.9	0.71
	(mean ± sd)

Table 4 : Comparison of patient who completed the questionnaire fully or not

 4

	LUTS: Lower urinary tract symptoms
	Sd: standard deviation
	FIM: Functional Independence Measure score
		Completed	Not completed
		questionnaire	questionnaire	p
		n=59	n=14
	Gender
	Male	11 (19%)	4 (29%)	0.46
	Female	48 (81%)	10 (71%)
	Age (mean ± sd)	52.5 ± 14.4	44.2 ± 12.1	0.04
	Neurologic etiology of LUTS
	Central	33 (59%)	7 (54%)	0.31
	Peripheral	11 (20%)	5 (38%)
	PP test 15	42 (71%)	8 (57%)	0.19
	Motor and/or sensory disorders	29 (49%)	8 (57%)	0.55
	FIM (mean ± sd)	104 ± 12	104 ± 13	0.94
	Number of catheters seen (mean ± sd)	2.8 ± 1.1	2.9 ± 1.4	0.91


Alice Cuzzocrea

Anthony Scemama

Wim J Briels
email: w.j.briels@utwente.nl

Saverio Moroni
email: moroni@democritos.it

Claudia Filippi
email: c.filippi@utwente.nl

Variational principles in quantum Monte Carlo: the troubled story of variance minimization

We investigate the use of different variational principles in quantum Monte Carlo, namely energy and variance minimization, prompted by the interest in the robust and accurate estimate of electronic excited states. For two prototypical, challenging molecules, we readily reach the accuracy of the best available reference excitation energies using energy minimization in a state-specific or state-average fashion for states of different or equal symmetry, respectively. On the other hand, in variance minimization, where the use of suitable functionals is expected to target specific states regardless of the symmetry, we encounter severe problems for a variety of wave functions: as the variance converges, the energy drifts away from that of the selected state. This unexpected behavior is sometimes observed even when the target is the ground state, and generally prevents the robust estimate of total and excitation energies. We analyze this problem using a very simple wave function and infer that the optimization finds little or no barrier to escape from a local minimum or local plateau, eventually converging to the unique lowest-variance state instead of the target state. While the loss of the state of interest can be delayed and possibly avoided by reducing the statistical error of the gradient, for the full optimization of realistic wave functions, variance minimization with current functionals appears to be an impractical route.

I. INTRODUCTION

Light-induced processes are at the heart of a variety of phenomena and applications which range from harnessing the response to light of biological systems to improving the technologies for renewable energies. The contribution of electronic structure theory in this field hinges on its ability to efficiently and accurately compute excited-state properties. In this context, the use of quantum Monte Carlo (QMC) methods is relatively recent and quite promising [1][2][3][4][5][6][7][8][9]: QMC approaches provide an accurate (stochastic) solution of the Schrödinger equation and benefit from a favorable scaling with system size and great ease of parallelization [10][11][12]. Importantly, recent methodological advancements [13][14][15][16] enable the fast calculation of energy derivatives and the optimization of many thousands of parameters for the internally consistent computation of QMC wave functions and geometries in the ground and excited states [9,17].

Here, we investigate the use of two different variational principles for ground and excited states in QMC, namely, variance and energy minimization, to assess whether they allow us to fully capitalize on the increased power of minimization algorithms and availability of accurate wave functions. Variance minimization techniques [18][19][20][21][22] have been extensively employed in QMC for the last 30 years but their potential for the computation of excited states has only recently been revisited and exploited to compute vertical excitation energies of various small molecules [23,24]. Different functionals for the optimization of the variance [19,22,23] have also been put forward with the common attractive feature of the builtin possibility to target a specific state and avoid in principle the complications encountered in energy minimization where, without constraints, one would generally collapse to lowerenergy states.

For our study, we select two molecules, a small cyanine dye and a retinal model, because of the difficulties they pose in the computation of the lowest vertical excitation energy [4,[25][26][27][28], and the different requirements in the procedure adopted in energy minimization: while the ground and excited states of the cyanine belong to different symmetries and can therefore be treated in a state-specific manner, this is not the case for the retinal model, where energy minimization must be performed in a state-average fashion. For both molecules and therefore regardless of the nature of the optimization, we find that energy minimization leads to the stable and fast convergence of the total energies of the states of interest. Furthermore, with the use of compact and balanced energy-minimized wave functions constructed through a selected configuration interaction (CI) approach, we recover vertical excitation energies which are already at the variational Monte Carlo (VMC) level within chemical accuracy (about 0.04 eV) of the reference coupled cluster or extrapolated CI values. On the other hand, for both molecules and for nearly all wave functions investigated, the optimization of all parameters in variance minimization is problematic since it results in the apparent loss of the state of interest over sufficiently long optimization runs, precluding the estimate of the excitation energy. This occurs for the different functionals originally proposed to stabilize the optimization and, surprisingly, in some cases also when targeting the ground state. This finding is unexpected, especially considering that variance minimization has been the method of choice in QMC for decades and is still routinely used, albeit for wave functions with a small number of parameters or where the optimization is limited either to few optimization steps or to the Jastrow factor.

To understand these newly-found issues, we examine how variance minimization behaves when optimizing the linear coefficients of a very simple wave function. Working in the linear sub-space spanned by a few approximate eigenvectors, we discover that the optimization of the CI parameters in variance minimization does not converge to the target eigenstate but to a different one: for approximate wave functions, the multiple minima of the variance have generally different values and, during the minimization, the system slowly reaches the eigenstate corresponding to the absolute minimum of the variance, no matter what the starting state is. We show that a similar behavior can be inferred also for more complicated wave functions and, while the process can be slowed down by reducing the statistical error on the gradient of the variance driving the minimization, it severely limits the use of variance minimization for the optimization of realistic wave functions.

In Section II, we recap the equations used for energy and variance optimization, discuss the procedure employed for the state-average case, and introduce the ingredients for a stable version of the Newton method in variance minimization. In Section III, we summarize the computational details and, in Section IV, present the accurate vertical excitation energies obtained in energy minimization and the difficulties encountered in variance minimization for both molecules. We elucidate these findings and conclude in Section V.

II. METHODS

We briefly introduce below the variance and energy minimization approaches used to optimize the wave functions in variational Monte Carlo. While we employ variance minimization as a state-specific approach to target a given state, we must distinguish between a state-specific and a state-average route for energy optimization when the excited state of interest is of different or equal symmetry, respectively, than other lower-lying states.

A. Wave function form

The wave functions employed in this work are of the Jastrow-Slater type, namely, the product of a determinantal expansion and a Jastrow correlation function, J , as

Ψ = J N det i=1 c i D i , (1)
where the determinants are expressed on single-particle orbitals and the Jastrow factor includes an explicit dependence on the electron-electron distances. Here, the Jastrow factor is chosen to include electron-electron and electron-nucleus correlation terms [START_REF]of two fifth-order polynomials of the electron-nuclear and the electronelectron distances, respectively, and rescale the inter-particle distances as R = (1 -exp(-κr))[END_REF]. For the determinantal component, we select the relevant determinants according to different recipes: i) very simple ansatzes such as Hartee-Fock (HF) or a CI singles (CIS) expansion recently put forward as a computationally cheap and sufficiently accurate wave function for excited states in QMC [8,[START_REF] Neuscamman | [END_REF]; ii) complete-active-space (CAS) expansions where small sets of important active orbitals are manually identified; iii) CI perturbatively selected iteratively (CIPSI) expansions generated to yield automatically balanced multiple states. All expansions are expressed in terms of spin-adapted configuration state functions (CSF) to reduce the number of variational parameters.

B. Energy minimization

For state-specific optimization in energy minimization, we employ the stochastic reconfiguration (SR) method [14,31] in a low-memory conjugate-gradient implementation [14]. Given a starting wave function Ψ depending on a set of parameters p, we denote the derivatives of Ψ with respect to a parameter p i as Ψ i = ∂ i Ψ. At every step of the SR optimization, the parameter variations, ∆p i , are computed according to the equation:

S∆p = - τ 2 g , (2)
where τ is a positive quantity chosen small enough to guarantee the convergence. The vector g is the gradient of the energy with components:

g i = ∂E ∂p i = 2 Ψ i | Ĥ|Ψ Ψ|Ψ -E Ψ|Ψ i Ψ|Ψ = 2 Ψ i Ψ E L -E L Ψ i Ψ , (3)
where E L = ĤΨ/Ψ is the so-called local energy and . denotes the Monte Carlo average of the quantity in brackets over the electron configurations sampled from Ψ 2 / Ψ|Ψ . The matrix S has components:

Sij = Ψ i |Ψ j Ψ|Ψ - Ψ|Ψ i Ψ|Ψ Ψ|Ψ j Ψ|Ψ = Ψ i Ψ Ψ j Ψ - Ψ i Ψ Ψ j Ψ ≡ Ψi Ψ Ψj Ψ , (4)
which is expressed in the last equality as the overlap matrix in the semi-orthogonal basis, Ψi = Ψ i -[ Ψ|Ψ i / Ψ|Ψ ]Ψ.

When the state of interest is energetically not the lowest in its symmetry class, we start from a set of wave functions for the multiple states which share the same Jastrow factor and orbitals but are characterized by different linear CI coefficients as

Ψ I = J N det i=1 c I i D i , (5)
where the superscript I indicates a particular state. To obtain a balanced description of the states of interest, we optimize the non-linear parameters of the orbitals and the Jastrow factor by minimizing the state-average energy [1]:

E SA = I w I Ψ I | Ĥ|Ψ I Ψ I |Ψ I , (6)
where the weights w I are kept fixed and I w I = 1. To this aim, we follow the SR scheme (Eq. 2) and use the gradient of the state-average energy

g SA i = I w I g I i , (7)
where g I i is the gradient with respect to a parameter p i of the energy of state I, which is computed from the wave function Ψ I and its derivatives as in Eq. 3. Moreover, in analogy to the single-state optimization, we introduce a weightedaverage overlap matrix defined as

SSA ij = I w I SI ij , (8)
where the overlap matrix for each state is computed from the corresponding wave function as in Eq. 4. We stress that, while the state-average SR procedure is defined simply by analogy with the single-state case, it does lead to the minimization of the state-average energy since we employ the appropriate g SA .

We alternate a number of optimization steps of the nonlinear parameters with the optimization of the linear coefficients c I i , whose optimal values are the solution of the generalized eigenvalue equations

H CI c I = E I S CI c I , (9)
where the Hamiltonian and overlap matrix elements are defined in the basis of the functions {J D i } and estimated through Monte Carlo sampling. After diagonalization of Eq. 9, orthogonality between the individual states is automatically enforced. To solve the eigenvalue equation with a memory efficient algorithm, we use the Davidson diagonalization scheme in which the lowest energy eigenvalues are computed without the explicit construction of the entire Hamiltonian and overlap matrices [14]. A similar procedure was recently followed in Ref. [32].

C. Variance minimization

To perform variance minimization, we can directly minimize the variance of the state of interest,

σ 2 = Ψ|( Ĥ -E) 2 |Ψ Ψ|Ψ , (10)
or follow a somewhat more stable optimization procedure by minimizing the expression

σ 2 ω = Ψ|( Ĥ -ω) 2 |Ψ Ψ|Ψ , (11)
where the energy ω is fixed during the optimization step and then appropriately modified to follow the current value of the energy as originally proposed in Ref. [19]. Recently, a functional Ω has been put forward,

Ω = Ψ|(ω -Ĥ)|Ψ Ψ|(ω -Ĥ) 2 |Ψ , ( 12
)
whose minimization is equivalent to variance minimization if ω is eventually updated to the running value of E -σ [23].

Because of its simplicity, we choose here the functional σ 2 ω but also compare the convergence behavior obtained with the functional Ω. To this aim, we use the Newton optimization method as in Ref. [22] and update the parameters as

∆p = -τ h -1 g , ( 13
)
where g is here the gradient of σ 2 ω and h its Hessian matrix, and the parameter τ is introduced to damp the size of the variations.

The components of the gradient are given by

g i = 2 ĤΨ i Ψ E L - Ψ i Ψ E 2 L ( 14
)
-ω

Ψ i Ψ E L + ĤΨ i Ψ -2 Ψ i Ψ E L ,
and we discuss other possible equivalent expressions and their relative fluctuations in the SI. The Hessian matrix elements require the second derivatives of the wave function and, to avoid their computation, we follow the same approximation strategy of the Levenberg-Marquardt algorithm [START_REF] William | Numerical Recepies Third Edition[END_REF] and manipulate the expression of the variance in a somewhat different way than proposed in Refs. [20,22,[START_REF] Toulouse | [END_REF], to obtain the approximate expression of the Hessian matrix

h ij = ∂ i E L + (E L -ω) Ψ i Ψ - Ψ i Ψ (15)
× ∂ j E L + (E L -ω) Ψ j Ψ - Ψ j Ψ ,
Details of the derivation and alternative expressions for the Hessian are given in the SI. We use the Newton method and the Hessian h (Eq. 15) when optimizing both σ 2 ω and the Ω functional in combination with the corresponding gradient. Furthermore, we follow Ref. [23] in keeping ω fixed to an appropriate guess energy for an initial number of minimization steps, upgrading it linearly to the running energy (or E -σ in the case of Ω) over some intermediate iteration steps, and then setting it equal to the current energy estimate for the rest of the run.

III. COMPUTATIONAL DETAILS

All QMC calculations are carried out with the program package CHAMP [START_REF]CHAMP is a quantum Monte Carlo program package[END_REF]. We employ scalar-relativistic energyconsistent HF pseudopotentials and the correlation-consistent Gaussian basis sets specifically constructed for these pseudopotentials [START_REF] Burkatzki | [END_REF][START_REF]For the hydrogen atom, we use a more accurate BFD pseudopotential and basis set[END_REF]. Unless otherwise specified, we use a double-ζ basis set minimally augmented with s and p diffuse functions on the heavy atoms and denoted here as maug-cc-pVDZ. Basis-set convergence tests are performed with the fully augmented double (aug-cc-pvDZ) and triple (aug-cc-pvTZ) basis sets. In all cases, the exponents of the diffuse functions are taken from the corresponding all-electron Dunning's correlation-consistent basis sets [START_REF] Kendall | [END_REF].

In the state-specific (energy and variance) optimization runs, we sample a guiding wave function that differs from the current wave function close to the nodes [39] to guarantee finite variances of the estimators of the gradient, overlap, and Hessian matrix elements. In the state-average energy minimizations, we employ equal weights for the multiple states and sample a guiding wave function constructed as Ψ 2 g = I |Ψ I | 2 , to ensure that the distribution sampled has a large overlap with all states of interest [1]. All wave function parameters (Jastrow, orbital, and CI coefficients) are optimized and the damping factor, τ , in the SR and the Newton method is set to 0.05 and 0.1, respectively, unless otherwise specified. In the DMC calculations, we treat the pseudopotentials beyond the locality approximation using the T-move algorithm [40] and employ an imaginary time-step of 0.05 a.u. which yields excitation energies converged to better than 0.01 eV as shown in the SI.

The HF, CIS, and complete-active-space self-consistentfield (CASSCF) calculations are carried out with the program GAMESS(US) [41,[START_REF] Gordon | Theory and applications of computational chemistry[END_REF]. For the cyanine dye, we consider different CAS expansions: a CAS(6,5) and CAS(6,10) correlating 6 π electrons in the orbitals constructed from the 2p z and 3p z atomic orbitals; a truncated CAS(14,13) consisting of 6 π and 8 σ electrons in 13 bonding and antibonding orbitals. For the retinal model, we employ a minimal CAS(6,6) active space of 6 π electrons in the orbitals constructed from the 2p z atomic orbitals.

The CIPSI calculations are performed with Quantum Package [START_REF] Garniron | [END_REF] and the determinantal expansions are constructed to be eigenstates of Ŝ2 . For the cyanine dye where ground and excited states have different symmetry, we follow two paths to construct the CIPSI expansions: i) we perform separate expansions for the two states starting from the corresponding CASSCF(6,10) orbitals, and match the variances of the CI wave functions to obtain a balanced description of the states. We find that this procedure leads to an automatic match of the second-order perturbation theory (PT2) energy contributions as shown in Table S1. ii) We perform the expansion of the two states simultaneously, using a common set of orbitals (the excited-state CASSCF(6,10) orbitals), and obtain automatically matched PT2 energy corrections during the expansion [9]. For the retinal model where the ground and excited states have the same symmetry, we have only one set of orbitals for the CIPSI expansions. In this case, we perform a simultaneous expansion with a selection scheme that matches the CI variances and also attempts to balance the PT2 energy contributions of the two states (see SI) [44].

All total energies are computed on the PBE0/cc-pVQZ ground-state geometry of the cyanine [45] and retinal molecules. The DFT geometry optimization of the retinal model is performed with the program Gaussian [START_REF] Frisch | Gaussian09, revision a.02; gaussian[END_REF]. The coupled cluster results are obtained with Psi4 [START_REF] Parrish | [END_REF].

IV. RESULTS

We compute the lowest π → π * vertical excitation energy of the cyanine dye (C 3 H 3 (NH 2 ) +

2 ) and the minimal model of the retinal protonated Schiff base (C 5 H 6 NH + 2 ) depicted in Fig. 1 and denoted as CN5 and PSB3, respectively. As already mentioned, while being generally challenging for electronic structure methods [4,[25][26][27][28], these examples are representative of the two cases of a ground (S0) and an excited (S1) state of different (CN5) and equal (PSB3) symmetry. Correspondingly, the energy minimization scheme is state-specific for CN5 and state-average for PSB3, while variance minimization affords a state-specific optimization for both molecules, at least in principle.

Ground and excited states of different symmetry

In Table I, we list the ground-and excited-state energies, and corresponding excitation energies of CN5 computed in VMC and DMC with different wave functions optimized by (state-specific) energy minimization. The simplest case consists of a single determinant (HF) and a HOMO-LUMO (HL) two-determinant wave function for the ground and the excited state, respectively. We then consider configuration interaction singles (CIS) expansions, CAS expansions with increasing active spaces, and balanced CIPSI expansions with different choices of the starting orbitals, namely, independent sets for the two states (CIPSI-SS) or a common set of orbitals (CIPSI-B 1 ). The excitation energies are displayed in Fig. 2.

The general trend is a decrease of the excitation energy towards the extrapolated full CI (exFCI) and approximate coupled cluster singles, doubles and triples model (CC3) reference values for better wave functions. As an exception, when we move from the HF/HL to CIS wave functions, the VMC energies of both states decrease but the corresponding excitation energy becomes worse. With increasingly large CAS expansions, both the total and the excitation energies improve but the convergence is very slow. For all these wave functions, the DMC excitation energy is lower than the VMC value and becomes within 0.1 eV of the reference results for the largest active spaces with about 50,000 and 70,000 determinants for the ground and the excited state, respectively. By comparison, the errors of TDDFT and CASPT2 can be as large as 0.4 and -0.2 eV, respectively [4,45].

The quality of the results exhibits a further, dramatic improvement with the use of CIPSI expansions. The VMC and DMC energies obtained with the smallest CIPSI wave function are lower than the corresponding values obtained with the largest CAS considered here. Furthermore, constructing ground-and excited-state CIPSI expansions with similar PT2 corrections leads to a balanced description of both states and to VMC excitation energies which change very little with increasing expansion size, being irregularly scattered over a VMC and DMC excitation energies of CN5 calculated with different wave functions optimized in energy minimization. The exFCI/aug-cc-pVDZ [48] and CC3/aug-cc-pVTZ reference values are also shown. The approximate total number of determinants for the CIPSI-SS wave functions of the ground and excited states is indicated.

small energy range of 0.08 eV. Importantly, the DMC excitation energies are compatible with the VMC ones and in excellent agreement with the CC3 and exFCI values. Finally, employing two different sets of orbitals to generate the CIPSI expansions leads to marginal differences, namely, to DMC excitation energies of 4.856(8) and 4.882 (8) eV, which are both bracketed by the reference values.

Having verified that state-specific energy optimization in combination with accurate wave functions allows the robust treatment of CN5, we now employ variance minimization with the σ 2 ω functional to optimize the CAS(6,5) and CAS (6,10) wave functions of the ground and excited states.

The convergence of the corresponding VMC variances and energies is shown in Fig. 3. For the smaller CAS(6,5), we observe that, while the variance converges rather quickly, the energy appears to do so more slowly and only after undershooting to a value which generally depends on the statistical error and initial conditions of the run. As reported in Table II, the optimal ground-and excited-state energies are higher by about 30 mHartree than the corresponding values obtained in energy minimization but the resulting excitation energy is compatible within statistical error.

If we move to the larger CAS(6,10) determinantal expansion, we find however that, while the variance reaches a stable value and the ground-state energy has a similar behavior to the CAS(6,5) case, the energy of the excited state grows steadily and it is therefore not possible to estimate the vertical excitation energy of the system. Surprisingly, even in the simplest case of the one-configuration (HF/HL) wave functions, the energy of the excited state keeps slowly rising even after 600 iterations as shown in Fig. 4, while the ground-state energy behaves similarly to the corresponding CAS cases.

We stress that, for most wave functions of Table I, we observe a steady growth of the excited-state energy in variance minimization and, therefore, wonder whether optimizations which appear converged are not simply affected by a much slower drift in the energy, which would become evident only in longer runs. Importantly, the apparently unstable behavior is independent of the initial value of ω and the number of steps over which we keep ω fixed (see Section S6). The use of a smaller or larger damping factors (i.e. τ = 0.04 and 0.2) leads to the same pathological growth of the excited-state energy, characterized by the same slope as a function of time as shown in Fig. S4. Moreover, we recover the same behavior also when using a gradient-only-based optimizer (see Fig. S7). Finally, minimizing the Ω functional instead of σ 2 ω yields an excited-state energy which ultimately rises with iterations as shown for the excited-state HL wave function in Fig. 4.

Ground and excited states of the same symmetry

For PSB3, we optimize the wave functions in energy minimization in a state-average fashion and report the resulting VMC and DMC total energies and vertical excitation energies in Table III. As in the CN5 case, CIPSI wave functions are superior to CAS expansions of similar size and, with only about 400 determinants, the use of CIPSI yields not only lower total energies but also a VMC vertical excitation energy in good agreement with the CC3 reference, largely correcting the error of 0.25 eV obtained with the CAS (6,6) wave function. For all CIPSI expansions, the DMC excitation energies are always quite close to the correspondent VMC results and, for the larger expansions, within 0.05 eV of the CC3 value. When we perform state-specific variance minimization, we encounter great difficulties in the convergence of the energies as we show for the HF/HL and CAS (6,6) wave functions in Fig. 5. Differently from CN5, we find in general that not only the energy of the excited state but also that of the ground state grows steadily with iteration number. We only obtain a seemingly stable energy for the HF wave function.

V. DISCUSSION AND CONCLUSIONS

While our results confirm the high accuracy reachable in QMC with energy minimization, they evidence severe problems in variance minimization which, in most cases, preclude the estimation of the excitation energy. To gain a better understanding of the troublesome behavior of the energy during variance minimization, we further investigate the simple case of the HL wave function of CN5 (Fig. 4) and find that the energy of the state drifts to higher values during variance minimization also when one optimizes only the LUMO orbital. Therefore, since optimization of an orbital can be achieved by mixing it with the unoccupied ones of the same symmetry, we can recast the LUMO optimization into the linear variation of the CI coefficients of the single excitations out of the LUMO orbital, which amount to only twelve additional CSFs in our basis set. With such a small expansion, we can then diagonalize the Hamiltonian in the basis of the CSFs times the Jastrow factor to estimate its thirteen eigenvalues and eigenvectors, and work directly in the basis of the eigenstates to assess the behavior of variance minimization when starting from the states which are optimal for energy minimization. The damping factor used in the Newton method is τ = 0.2.

In Fig. 6, we show the evolution of the VMC variance and energy for four variance minimization runs in which we start from different eigenvectors, taking the corresponding eigenvalues as initial target energies ω. In particular, we consider the lowest state in B 1 symmetry as well as the second, fourth, and thirteenth (corresponding to the highest energy) states. We note that, since our states are not exact eigenstates of the full Hamiltonian, the corresponding variances of the local energy are non zero and are spread over about 0.5 a.u. with the lowest value in correspondence of the second state. In principle, one would expect to find a feature of the variance landscape -ideally a local minimum-near each of the approximate eigenstates since the functionals σ 2 ω or Ω are designed to select a particular state through the initial value of ω, and minimize the variance of that state. Here, the selection of the state is further facilitated starting each run precisely from the chosen eigenstate, and variance minimization should perform minor adjustments of the initial parameters from their optimal values for the energy.

The behavior illustrated in Figs. 6 is totally different, with all optimization runs leaking down to successive lowervariance states and eventually converging to the absolute minimum corresponding to the second eigenstate. The staircase shape of the variance evolution points to the presence of flat regions of the variance landscape close to the eigenstates, from which the optimization can eventually escape. This is further corroborated if we follow the evolution of the CI coefficients as shown starting from the highest-energy state in Fig. 7: the initial coefficient quickly decreases to zero and other eigenstates become populated until convergence on the second state. In proximity of some the variance displays a more pronounced plateau, where the system spends enough time to acquire the full character of that particular state. It is also interesting to note that the states are populated sequentially with the order determined by decreasing energies. We stress that we observe a similar behavior of the variance also when using the Ω functional starting from the same set of approximate eigenstates (see Fig. S8).

In Fig. 8, we investigate the impact of the statistical error on the loss of the selected state. In particular, we focus on the evolution of the variance and the energy starting from the 4 th eigenvector for different lengths of the VMC runs used to compute the gradient and Hessian matrix. The shortest run (larger statistical error) looses the target state in a slightly smaller number of steps. However the intermediate and the longest run give very similar results, suggesting that even longer VMC runs would not stabilize the target state.

Finally, to verify that the understanding gained here also applies to more complicated wave functions, we revisit the very problematic optimization of the excited-state CAS (6,10) wave function (Fig. 3) and perform a much longer calculation, finding that the energy eventually converges as shown in Energy FIG. 7. Evolution of the square of the CI coefficients c 2 i (offset by i for clarity) of the small expansion of CN5 during variance minimization, for the run starting from the 13 th eigenvector; in the inset, the evolution of the energy is replicated to emphasize flat regions in the energy landscape close to an eigenstate (i.e. when the corresponding ci ∼ 1). Fig. 9. For the final set of Jastrow and orbital parameters, we determine the eigenvalues in the linear space of the determinants times the Jastrow factor and recover a similar behavior to what observed in the simple example: the minimization of σ 2 ω brings the system approximately to an eigenstate with a lower variance, which is in this case the 4 th one.

In summary, we have shown that the combination of energy minimization with an appropriate choice of the groundand excited-state wave functions via a balanced CIPSI procedure leads to excitation energies that are in excellent agreement already at the VMC level with the reference values. In particular, we obtained a robust convergence of the total ground-and excited-state energies, and a very accurate excitation energy not only in the easier state-specific case of CN5 but also when employing energy minimization in a stateaverage fashion for PSB3. On the other hand, we encountered severe problems when employing variance minimization since, over sufficiently long optimization runs, one may loose the state of interest in favor of a state with lower variance, as we clearly demonstrated with a simple but realistic example. Even though, theoretically, the functionals σ 2 ω and Ω have a built-in possibility to target the energy of a specific state, in practice, this is generally not sufficient to maintain the parameters close to the desired local minimum of the variance. Therefore, these considerations lead to the conclusion that, with the present functionals and no a priori knowledge of the parameter landscape of the variance for the system of interest, energy minimization is a safer and more stable procedure.

FIG. 1 .

 1 FIG. 1. Schematic representations of the CN5 (left) and PSB3 (right) molecules. White, gray, and blue denote hydrogen, carbon, and nitrogen, respectively.

FIG. 4 .

 4 FIG. 4. Convergence of VMC energy of the ground (left) and excited (right) states of CN5 in the optimization of the HF/HL wave functions within variance minimization with the σ 2 ω (our default) and the Ω functional.

FIG. 5 .

 5 FIG.5. Convergence of the VMC energy of the ground (red) and excited (blue) states of PSB3 in the optimization of the RHF/HL and CAS(6,6) wave functions within variance minimization.

FIG. 8 .FIG. 9 .

 89 FIG.8. Convergence of the variance (left) and energy (right) for different lengths of the Monte Carlo runs used to compute the gradient and Hessian matrix during the optimization, starting from the 4 th eigenvector. NMC is the number of Monte Carlo steps used in Fig.6.

TABLE I .

 I VMC and DMC total energies (a.u.) and excitation energies (∆E, eV) of CN5 obtained for different wave functions optimizing all parameters (Jastrow, orbital, and CI coefficients) in energy minimization.

			WF	No. det	No. parm	EVMC	∆EVMC	EDMC	∆EDMC
				S0	S1	S0	S1	S0	S1	S0	S1
			HF/HL	1	2	516 529 -40.8372(4) -40.6460(3) 5.202(14) -40.9378(3) -40.7509(3) 5.086(11)
			HF/CIS	1	980 516 4751 -40.8372(4) -40.6505(3) 5.080(14) -40.9378(3) -40.7533(3) 5.020(11)
			CIS	999 980 5260 4751 -40.8444(4) -40.6505(3) 5.278(14) -40.9393(3) -40.7533(3) 5.061(11)
			CAS(6,5)	52	48 567 561 -40.8468(4) -40.6583(4) 5.130(15) -40.9433(3) -40.7582(2) 5.038(10)
			CAS(6,10) 7232 7168 3134 3064 -40.8498(4) -40.6628(4) 5.090(15) -40.9439(3) -40.7594(3) 5.022(11)
			CAS(14,13) 48206 72732 9480 11727 -40.8583(3) -40.6713(3) 5.091(10) -40.9442(7) -40.7611(7) 4.983(26)
			CIPSI-SS	376 1094 1567 2609 -40.8646(3) -40.6842(3) 4.908(12) -40.9467(3) -40.7665(3) 4.905(10)
				1344 4382 2478 4531 -40.8798(3) -40.7013(3) 4.857(13) -40.9502(2) -40.7711(2) 4.872(09)
				2460 8782 3555 6561 -40.8896(3) -40.7099(3) 4.890(12) -40.9532(2) -40.7748(2) 4.856(09)
				3913 14114 4842 8312 -40.8941(2) -40.7167(3) 4.828(11) -40.9559(2) -40.7775(2) 4.856(08)
			CIPSI-B1	2456 6120 3971 5466 -40.8847(2) -40.7053(2) 4.880(09) -40.9521(2) -40.7727(2) 4.881(09)
				4829 13130 5737 8021 -40.8945(3) -40.7150(3) 4.889(13) -40.9560(2) -40.7766(2) 4.882(08)
			exFCI/aug-cc-pVDZ [48]				4.89
			CC3/aug-cc-pVDZ					4.851
			CC3/aug-cc-pVTZ					4.844
		5.3					VMC
							DMC
	Vertical excitation energy (eV)	4.9 5.0 5.1 5.2	exFCI CC3
		4.8
				CAS			CIPSI
			HF/HL CIS (6,5) (6,10) (14,13) 1k	6k	11k 18k
	FIG. 2.

TABLE II .

 II VMC energies and variances (a.u.) and vertical excitation energies (eV) of CN5 obtained with energy and variance minimization.

					Energy min.	Variance min.
			E(S0)		E(S1)	∆E σ 2 (S0) σ 2 (S1)	E(S0)	E(S1)	∆E σ 2 (S0) σ 2 (S1)
			CAS(6,5) -40.8468(4) -40.6583(4) 5.13(1) 0.928 0.940 -40.8170(5) -40.6270(5) 5.17(2) 0.856 0.862
			CAS(6,10) -40.8498(4) -40.6628(4) 5.09(1) 0.928 0.930 -40.8163(4)	-	-	0.855	-
		-40.79	Ground-state energy	-40.53	Excited-state energy
	Energy (a.u.)	-40.82 -40.81 -40.80		-40.62 -40.59 -40.56
		-40.83
		1.00	Ground-state variance	1.04	Excited-state variance
	Variance (a.u.)	0.88 0.92 0.96	CAS(6,5) CAS(6,10)	0.88 0.92 0.96 1.00		CAS(6,5) CAS(6,10)
		0.84	0 100 200 300 400 500 600 700	0.84	0 100 200 300 400 500 600 700
			Optimization iteration		Optimization iteration

FIG. 3. Convergence of the VMC energy (top) and variance (bottom)

of the ground (left) and excited (right) states of CN5 in the optimization of the CAS(6,5) and CAS

(6,10)

wave functions in variance minimization.

TABLE III .

 III VMC and DMC total energies (a.u.) and vertical excitation energies (eV) of PSB3 obtained with different wave functions optimized in energy minimization. The horizontal lines in the energy plot correspond to the eigenvalues in this reduced space, and the colored ones are the eigenstates used as starting point in four optimization runs.

			WF	No. det No. param	VMC	DMC
							E(S0)	E(S1)	∆E	E(S0)	E(S1)	∆E
			CAS(6,6) 400	1645	-42.8091(2) -42.6471(2) 4.409(9) -42.9118(2) -42.7541(2) 4.293(6)
			CIPSI	422	4011	-42.8174(2) -42.6623(2) 4.221(9) -42.9133(2) -42.7578(2) 4.233(6)
				1158	5968	-42.8297(2) -42.6735(2) 4.252(9) -42.9160(2) -42.7609(2) 4.221(6)
				2579	8106	-42.8357(2) -42.6796(2) 4.247(9) -42.9169(2) -42.7621(2) 4.214(6)
			CC3/aug-cc-pVDZ			4.19
			CC3/aug-cc-pVTZ			4.16
		1.4 1.5	Eig 13 Eig 4			Eig 2 Eig 1
	Variance (a.u.)	1.1 1.2 1.3		0.98 1.00
					0	250	500	750
		1.0
		-38.5
	Energy (a.u.)	-39.5 -39.0
		-40.0
		-40.5
		0	400 Optimization iteration 800 1200	1600
	FIG. 6. Convergence of the VMC variance (top) and energy (bottom)
	of CN5 in the CI optimization of a small expansion (see text) with
	variance minimization.

ACKNOWLEDGMENT

A.C. is supported by the "Computational Science for Energy Research and Netherlands eScience Center joint program" (project CSER.JCER.022) of the Netherlands Organisation for Scientific Research (NWO). This work was carried out on the Dutch national supercomputer Cartesius with the support of SURF Cooperative.

CONTENT OF SI

Derivation and discussion of the expressions of the gradient and approximate Hessian of the variance; CIPSI energies for various expansions; basis-set dependence of the VMC and DMC excitation energies; DMC excitation energy versus time step; dependence of variance minimization on the choice of ω, number of steps with ω fixed, damping factor in the Newton method, and statistical error; optimizations with a gradientbased optimizer and with the Ω functional.