The cold weather has finally ended and it's time to put away those hoodies and sweatpants and get into the spring spirit. We all love to cuddle up with cute beanies and fuzzy jackets but now we need to take off some of those layers and enjoy the nice weather with fashionable skirts, booties and t-shirts.
You can finally take out your favorite tees that match with just about any pair of jeans. The more color the better when it comes to comfortable fitting t-shirts. You can dress up any tee with booties or even dress it down with casual shorts and sandals. The different colors and patterns on new tees, popular in stores, allow for them to be easily paired with anything your closet has to offer. Tees and shorts are great because they are comfortable and affordable.
Casual tees can also be dressed up with a simple or colorful patterned mini skirt. If the weather is being unpredictable and you don't want to wear pants, but also don't want to wear shorts, throw on a casual, cute skirt with various buttons all the way down. This simple piece of clothing will make your outfit stand out. A skirt can be casual for a movie date or fancy for a wedding depending on how you dress it up. Mini skirts can be made fancy with a nice blouse and heels or booties, but can be more casual with sneakers and a comfortable t-shirt. Even on days where the weather is chilly, one can always throw on a turtleneck sweater with a faux-leather skirt.
Girls aren't the only ones who can get into spring fashion, men can too. Instead of the basic long sleeve t-shirt and jeans, they can dress their outfit up with a simple, cute short sleeve button up. Button ups can be comfortable and cool with hot days. These shirts can also be in various colors and patterns making them easy to match just about anything a man has in his closet. Not only can they be worn on casual days with jeans or khakis but can also be dressed up with a nice pair of slacks and shoes.


Four holiday cottages on a working Dairy Farm close to Tobermory. All have been recently re-furbished to a very comfortable standard. They are heated from the farm's log burning boiler and also have their own wood burning stoves.
Kitchen: Electric cooker - fridge freezer - dishwasher - microwave - toaster - electric kettle - cafetiere - washing machine, tumble drier and iron in shared utility.
The farm properties have shared laundry facilities and use of the farm's heated indoor swimming pool and sauna. Guests should bring their own swimming towels & flotation equipment if required, children must be supervised by adults at all times.


The Rambler is an affordable, reliable magnetic stripe card reader that transforms smartphones and tablets into secure, fully functioning mPOS solutions for a seamless user experience.
The Rambler device employs AnywhereCommerce's patented audio jack connectivity and state-of the-art encryption technology to ensure secure, reliable operation in mobile POS applications. Now businesses have the freedom to process credit card transactions wherever they need to serve customers.
The Rambler is ideally suited for any business seeking an affordable payment solution for on-site credit card processing. Merchants using Rambler can realize lower cost transactions by taking advantage of card-present processing rates. Additionally, Rambler optimizes cash flow and accounts receivables by cutting delays in billings and collections, common amongst home service providers. Rambler is also well suited for enterprise applications and reduces PCI scope and merchants' overall risk profile.


OTTAWA -- Indigenous women are probably not the only ones who have been coerced into being sterilized without wanting to be, an assistant deputy minister at Health Canada told the House of Commons' health committee Thursday.
Abby Hoffman stressed that she did not want to suggest anything about the scale of the issue, but also said it's not hard to imagine a woman with an addiction or a mental illness agreeing to be sterilized after being presented with the option in a way that didn't get her genuine informed consent.
"At the end of the day what may have seemed to the provider as consent might not in reality have been consent as one would think consent should be managed," Hoffman testified.
Hoffman -- a trailblazer in women's sport and four-time Olympian who is now in charge of Health Canada's strategic policy branch -- joined officials from three departments to testify before the health committee over recent reports on Indigenous women who say they were pressured during childbirth into having tubal ligations.
He said in an interview he believes Canada is at the "very cusp" of uncovering serious human-rights abuses affecting women who are vulnerable for numerous reasons, including those who are Indigenous, those who live with disabilities and those who have low incomes.
"There are very few things that are as profoundly a part of a person's individual liberty as control over their reproduction and the choice of whether or not to have children," he said. "When that is taken away from someone in a very vulnerable position, without their consent, permanently, I think that's a violation of a particularly fundamental and heinous nature."
In December, Health Minister Ginette Petitpas Taylor and then Indigenous-services minister Jane Philpott wrote a letter to the provinces, territories and members of the medical community proposing a working group to examine cultural competence in health -- the attitudes and skills that doctors and nurses and other medical workers use to work with people from different backgrounds -- while they also cited reports of women being sterilized against their will.
The ministers' letter went out after the United Nations Committee Against Torture told Canada to stop the "extensive forced or coerced sterilization" of Indigenous women and girls -- a finding that prompted calls for federal action by human-rights groups and Davies' New Democrats.
"We know this is a violation of the right to health and a woman's right to receive consent-based services," she said.
"That's why a study or a committee needs to be formed to further investigate this," she said.
Kennell's organization and Amnesty International Canada have been pushing for an immediate and longer-term study of sterilization without consent.
"Whoever within Parliament is able to do that study, I think it will be welcomed," said Jackie Hansen, Amnesty's gender-rights campaigner.
"There is such a need to have a really deep look into how this practice has continued and what needs to be done to stop it and what sort of justice the women impacted and their families want to see."


text:
Didrick teamed up with Owl City's Adam Young For Debut Single on Monstercat [LISTEN]
Didrick and the 'Fireflies' singer are 'Ready to Fly' with a new single.
Miljan Milekic
The young gun from the Monstercat family, Didrick, has teamed up with Adam Young, best known under his moniker Owl City, for a collaboration track "Ready to Fly." The song is out via brand new imprint Monstercat: Instinct.
Swedish DJ, producer, and musician Didrick, who was also responsible for Monstercat's 3-year and 5-year anniversary mixes, has finally released his first single. The future bass track "Ready to Fly" is the first ever release on the new imprint Monstercat: Instinct and features vocals from "Fireflies" singer and electro-pop star Adam Young. The song is also Young's first appearance on Monstercat.
Monstercat: Instinct is a brand new imprint from the Canadian label and focuses on pop-infused and more radio-friendly music. On the other side, Monstercat presented another new imprint, bass-heavy Monstercat: Uncaged.
"Monstercat: Instinct represents an important development in the tastes and interests of the Monstercat family. As we've grown, new music, events, and clothing brands inspire us to strive for further innovation. Through an exploration of art, creativity, and beauty. Instinct will develop the unique stories of our artists worldwide"
said Mike Darlington, CEO of Monstercat. He also added:
"Monstercat: Uncaged is the continued evolution of the music that defined our early years. The combination of live events, bass music, and gaming laid the groundwork for our deep love of electronic music. Uncaged allows us to heighten the focus on these core elements to create a compelling experience for our community."
If you're up for some quality production, beautiful and chill melodies, as well as haunting vocals, make sure to check the song below.
Monstercat UncagedDidrickMonstercatOwl CityMonstercat Instinct
Sign up for the latest news, exclusive giveaways & more.
Monstercat Unveils Two New Distinct Brands, Uncaged and Instinct
Monstercat teases us with two new brands!
Stephen Makes Monstercat Debut With "Stay" and "Play Me Like a Violin"
The popular dance label has released a pair of tracks from one of the most unique artists to ever be under their label.
KUURO Team Up With Monstercat For Bass Anthem "Omen"
Keep your eye on Kuuro.
Rootkit Returns to Monstercat With New Single "Taking Me Higher"
Out now on Monstercat Instinct Vol.2
Rogue New Single "Mirage" Out Now [Listen]
Monstercat is back with another future bass banger
Holly & Loosid Release New Single "Fire Flower" Out Now [Listen]
Out now on Monstercat Uncaged Vol. 6.
Dirtyphonics Talks New Bass-Heavy Collab, Monstercat, And More At Miami Music Week [Interview]
We caught up with the French group in Miami to talk about their new doomsday track with Bassnectar & Ragga Twins, Skateboarding, Trump, and much more.
Soulji Releases Debut EP on Monstercat
The enigmatic producer releases his first EP on the well known Canadian label

redpajama_set_name: RedPajamaCommonCrawl


As more and more health systems are swallowing up smaller systems, I have been getting contacted by physicians who are concerned that their employment agreement has "expired". For example, a physician's employment agreement may provide that the agreement ends on June 30, 20xx. It is becoming more common for that date to come and go with no new agreement offered. Physicians in this position are rightly concerned about their status.
The good news in this situation is that most physician employment agreements are "evergreen". That is, they automatically renew at the end of each term. Therefore, the physician's employment agreement will remain in full force and effect even though the "expiration date" has passed.
In the somewhat unusual case where the physician's employment agreement does not automatically renew, the physician should still be protected. Contract law generally provides that where both parties continue to perform under the agreement (i.e., the physician continues to provide services and the employer continues to pay compensation and benefits) the contract is considered "affirmed". Therefore, the contract will be treated as if it had not expired.
Although you may be relieved to know that the employer is still obligated to keep paying you, you should still keep track of the expiration date. I was recently contacted by a primary care physician whose three-year contract had expired a few months ago. She had an offer from a nearby competing health system of about $60,000 a year more salary. When I spoke to her current employer's lawyer about the situation, a new and much more lucrative contract was quickly forthcoming.
It is rarely a good idea to allow an old contract to continue. Your value to the employer may have increased significantly, either through changes to market compensation or because of your reputation among referring physicians. In addition, if you didn't have an experienced physician's contract lawyer review the first contract, there could be significant legal traps that can be removed or lessened in a new contract negotiation.


The histogram is a very useful way of checking and confirming exposure, especially out in the field. It may just look like a bunch of strange squigly lines or a series of jagged mountains, but the histogram tells you exactly what your photograph really looks like in terms of exposure and color. Since it is often hard to see the camera's review screen in direct sunlight to judge exposure, the histogram for a photographer is like a pilot flying an airplane by using only instruments. You do not even need to see the photograph to tell if it is a dark image, a bright one, or even a very colorful one. Essentially, the histogram looks like a rectangular box representing 256 shades of gray or individual colors, or all of them (see the histograms above showing grayscale and RGB color as Aperture 3 represents the histogram). Black is on the left edge and white is on the right edge. The more data to the left, means the darker an image is, and potentially underexposed. The more date to the right means a brighter image, and possibly an overexposed ones.
It is usually best to try and get as much of the data in the center forming a nice mountain, with no spikes at either the far left or far right edges. In the samples above, the left histogram has most of its data on the right side, indicating the photo is a bright exposure, as can be confirmed by looking at the photo itself. The histogram on the right is much more balanced and a more even exposure, as seen in the photo showing a bluer sky and better whites in the pelicans' heads.
The above image of Smacks Bayou (as seen from just off my back patio) has a histogram that is about as ideal as it can get. The majority of the data for grayscale and RGB color is in the middle-center and there is a lot of range in those colors. The taller the peaks, the more of that particular color or shade of gray. There are also no spikes at either far edge of the histogram.


The Grey Ghost Precision compensator was designed to minimize recoil and muzzle rise. Features a unique porting system to redirect the forward moving gasses rearward to greatly reduce recoil. Black nitride coating for durability and ease of cleaning. Easy installation with supplied crush washer.


text:
package de.cau.cs.kieler.klay.layered.p3order;

import java.util.List;
import java.util.ListIterator;
import java.util.Random;

import com.google.common.collect.HashMultimap;
import com.google.common.collect.Lists;
import com.google.common.collect.Multimap;

import de.cau.cs.kieler.core.alg.IKielerProgressMonitor;
import de.cau.cs.kieler.kiml.options.PortSide;
import de.cau.cs.kieler.klay.layered.ILayoutPhase;
import de.cau.cs.kieler.klay.layered.IntermediateProcessingConfiguration;
import de.cau.cs.kieler.klay.layered.graph.LEdge;
import de.cau.cs.kieler.klay.layered.graph.LGraph;
import de.cau.cs.kieler.klay.layered.graph.LNode;
import de.cau.cs.kieler.klay.layered.graph.LNode.NodeType;
import de.cau.cs.kieler.klay.layered.graph.LPort;
import de.cau.cs.kieler.klay.layered.graph.Layer;
import de.cau.cs.kieler.klay.layered.intermediate.IntermediateProcessorStrategy;
import de.cau.cs.kieler.klay.layered.properties.GraphProperties;
import de.cau.cs.kieler.klay.layered.properties.InternalProperties;
import de.cau.cs.kieler.klay.layered.properties.PortType;
import de.cau.cs.kieler.klay.layered.properties.Properties;

/**
 * Crossing minimization module that performs one or more sweeps over the layers while applying a
 * two-layer crossing minimization heuristic on each pair of layers. Inspired by
 *

 *
Kozo Sugiyama, Shojiro Tagawa, and Mitsuhiko Toda. Methods for visual understanding of
 * hierarchical system structures. IEEE Transactions on Systems, Man and Cybernetics, 11(2):109–125,
 * February 1981.
 *
Michael Forster. A fast and simple heuristic for constrained two-level crossing reduction. In
 * Graph Drawing, volume 3383 of LNCS, pp. 206-216. Springer, 2005.

 *

 *
This is the main layer sweep crossing minimization class that makes use of a number of further
 * classes. This one implements the actual layer sweep logic.

 *
 *

 *
Precondition:

 *
The graph has a proper layering, i.e. all long edges have been splitted; all nodes have at
 * least fixed port sides.

 *
Postcondition:

 *
The order of nodes in each layer and the order of ports in each node are optimized to yield
 * as few edge crossings as possible

 *

 *
 * @author msp
 * @author cds
 * @author ima
 * @kieler.design proposed by msp
 * @kieler.rating proposed yellow by msp
 */
public final class LayerSweepCrossingMinimizer implements ILayoutPhase {

    /** intermediate processing configuration. */
    private static final IntermediateProcessingConfiguration INTERMEDIATE_PROCESSING_CONFIGURATION =
            IntermediateProcessingConfiguration.createEmpty()
                    .addBeforePhase3(IntermediateProcessorStrategy.LONG_EDGE_SPLITTER)
                    .addBeforePhase4(IntermediateProcessorStrategy.PORT_DISTRIBUTER)
                    .addBeforePhase4(IntermediateProcessorStrategy.GREEDY_SWITCH)
                    .addBeforePhase4(IntermediateProcessorStrategy.IN_LAYER_CONSTRAINT_PROCESSOR)
                    .addAfterPhase5(IntermediateProcessorStrategy.LONG_EDGE_JOINER);

    /**
     * {@inheritDoc}
     */
    public IntermediateProcessingConfiguration getIntermediateProcessingConfiguration(
            final LGraph graph) {
        IntermediateProcessingConfiguration configuration =
                IntermediateProcessingConfiguration.fromExisting(INTERMEDIATE_PROCESSING_CONFIGURATION);

        if (graph.getProperty(InternalProperties.GRAPH_PROPERTIES).contains(
                GraphProperties.NON_FREE_PORTS)) {

            configuration.addBeforePhase3(IntermediateProcessorStrategy.PORT_LIST_SORTER);
        }

        return configuration;
    }

    /**
     * Array of port ranks used for sorting nodes and ports.
     */
    private float[] portRanks;
    /**
     * Complete node order of the best layer sweep.
     */
    private NodeGroup[][] bestSweep;
    /**
     * Complete node order of the current layer sweep.
     */
    private NodeGroup[][] curSweep;
    /**
     * Complete node order of the previous layer sweep.
     */
    private NodeGroup[][] prevSweep;
    /**
     * Crossings counter for normal edges.
     */
    private BarthJuengerMutzelCrossingsCounter normalCrossingsCounter;
    /**
     * Crossings counter for hyperedges.
     */
    private HyperedgeCrossingsCounter hyperedgeCrossingsCounter;
    /**
     * Both of the two previous crossing counters use the same
     * code to count in-layer crossings.
     */
    private AbstractCrossingsCounter inlayerCrossingsCounter;
    /**
     * Whether the layers contain hyperedges or not.
     */
    private boolean[] hasHyperedgesEast;
    private boolean[] hasHyperedgesWest;
    /**
     * Layout units represented by a single node.
     */
    private final Multimap<LNode, LNode> layoutUnits = HashMultimap.create();

    /**
     * Initialize all data for the layer sweep crossing minimizer.
     *
     * @param layeredGraph a layered graph
     */
    private void initialize(final LGraph layeredGraph) {
        int layerCount = layeredGraph.getLayers().size();

        // Remember the best, current and previous sweep; they basically save the node oder
        // per layer for the different sweeps of the algorithm
        bestSweep = new NodeGroup[layerCount][];
        curSweep = new NodeGroup[layerCount][];
        prevSweep = new NodeGroup[layerCount][];

        int[] inLayerEdgeCount = new int[layerCount];
        boolean[] hasNorthSouthPorts = new boolean[layerCount];
        hasHyperedgesEast = new boolean[layerCount];
        hasHyperedgesWest = new boolean[layerCount];

        int nodeCount = 0;
        int portCount = 0;

        // Iterate through the layers, initializing port and node IDs, collecting
        // the nodes into the current sweep and building the layout unit map
        ListIterator<Layer> layerIter = layeredGraph.getLayers().listIterator();
        while (layerIter.hasNext()) {
            Layer layer = layerIter.next();

            int layerIndex = layerIter.previousIndex();
            int layerNodeCount = layer.getNodes().size();
            // Empty layers are not allowed!
            assert layerNodeCount > 0;

            // Initialize this layer's node arrays in the different sweeps
            bestSweep[layerIndex] = new NodeGroup[layerNodeCount];
            prevSweep[layerIndex] = new NodeGroup[layerNodeCount];
            curSweep[layerIndex] = new NodeGroup[layerNodeCount];
            inLayerEdgeCount[layerIndex] = 0;
            hasNorthSouthPorts[layerIndex] = false;

            ListIterator<LNode> nodeIter = layer.getNodes().listIterator();
            while (nodeIter.hasNext()) {
                LNode node = nodeIter.next();

                // Create node group and register layout unit
                NodeGroup nodeGroup = new NodeGroup(node);
                curSweep[layerIndex][nodeIter.previousIndex()] = nodeGroup;
                node.id = nodeCount++;
                node.setProperty(InternalProperties.NODE_GROUP, nodeGroup);
                LNode layoutUnit = node.getProperty(InternalProperties.IN_LAYER_LAYOUT_UNIT);
                if (layoutUnit != null) {
                    layoutUnits.put(layoutUnit, node);
                }

                // Count in-layer edges
                for (LPort port : node.getPorts()) {
                    port.id = portCount++;
                    for (LEdge edge : port.getOutgoingEdges()) {
                        if (edge.getTarget().getNode().getLayer() == layer) {
                            inLayerEdgeCount[layerIndex]++;
                        }
                    }
                    if (port.getSide() == PortSide.EAST) {
                        if (port.getOutgoingEdges().size() + port.getIncomingEdges().size() > 1) {
                            hasHyperedgesEast[layerIndex] = true;
                        }
                    } else if (port.getSide() == PortSide.WEST) {
                        if (port.getOutgoingEdges().size() + port.getIncomingEdges().size() > 1) {
                            hasHyperedgesWest[layerIndex] = true;
                        }
                    } else {
                        // North and south port should not have any connected edges at this moment.
                        assert port.getOutgoingEdges().isEmpty() && port.getIncomingEdges().isEmpty();
                    }
                }

                // Count north/south dummy nodes
                if (node.getNodeType() == NodeType.NORTH_SOUTH_PORT) {
                    inLayerEdgeCount[layerIndex]++;
                    hasNorthSouthPorts[layerIndex] = true;
                }
            }
        }

        // Check whether every (or no) combination of layers, considering both
        //  forward and backward sweeps, involves hyperedges
        // If neither is the case, we need a counting algorithm for both
        //  hyperedges and straightline edges
        boolean allLayerCombinationsHaveHyperedges = true;
        boolean noLayerCombinationHasHyperedges = true;
        for (int i = 0; i < hasHyperedgesWest.length - 1; i++) {
            boolean b = hasHyperedgesEast[i] || hasHyperedgesWest[i + 1];
            allLayerCombinationsHaveHyperedges &= b;
            noLayerCombinationHasHyperedges &= !b;
        }

        // Initialize the port positions and ranks arrays
        portRanks = new float[portCount];
        int[] portPos = new int[portCount];

        // Create the crossings counter modules
        if (!allLayerCombinationsHaveHyperedges) {
            normalCrossingsCounter = new BarthJuengerMutzelCrossingsCounter(inLayerEdgeCount,
                    hasNorthSouthPorts, portPos);
            inlayerCrossingsCounter = normalCrossingsCounter;
        }

        if (!noLayerCombinationHasHyperedges) {
            hyperedgeCrossingsCounter = new HyperedgeCrossingsCounter(inLayerEdgeCount,
                    hasNorthSouthPorts, portPos);
            inlayerCrossingsCounter = hyperedgeCrossingsCounter;
        }
    }

    /**
     * Releases all created data so the GC can reap them.
     */
    private void dispose() {
        portRanks = null;
        bestSweep = null;
        curSweep = null;
        prevSweep = null;
        normalCrossingsCounter = null;
        hyperedgeCrossingsCounter = null;
        hasHyperedgesEast = null;
        hasHyperedgesWest = null;
        layoutUnits.clear();
    }

    /**
     * {@inheritDoc}
     */
    public void process(final LGraph layeredGraph, final IKielerProgressMonitor monitor) {
        monitor.begin("Layer sweep crossing minimization", 1);

        // Fetch the graph's randomizer.
        Random random = layeredGraph.getProperty(InternalProperties.RANDOM);

        // Find the number of layers. If there's only one, no crossing minimization is necessary.
        int layerCount = layeredGraph.getLayers().size();
        if (layerCount < 2) {
            monitor.done();
            return;
        }

        // Initialize the algorithm
        initialize(layeredGraph);
        int bestSweepCrossings = Integer.MAX_VALUE;

        // Determine the requested number of runs
        int runCount = layeredGraph.getProperty(Properties.THOROUGHNESS);

        // Initialize the compound graph layer crossing minimizer
        IConstraintResolver constraintResolver = new ForsterConstraintResolver(layoutUnits);
        ICrossingMinimizationHeuristic crossminHeuristic = new BarycenterHeuristic(constraintResolver,
                random, portRanks);

        // Create port distributors
        NodeRelativePortDistributor nodeRelativePortDistributor
                = new NodeRelativePortDistributor(portRanks);
        LayerTotalPortDistributor layerTotalPortDistributor
                = new LayerTotalPortDistributor(portRanks);
        AbstractPortDistributor portDistributor;

        // Perform the requested number of runs, each consisting of several sweeps
        for (int run = 0; run < runCount && bestSweepCrossings > 0; run++) {
            // Each run is randomly determined to be a forward or a backward run
            boolean forward = random.nextBoolean();
            int fixedLayerIndex = forward ? 0 : layerCount - 1;
            NodeGroup[] fixedLayer = curSweep[fixedLayerIndex];

            // Randomly choose a port distribution method for this run
            portDistributor = random.nextBoolean()
                    ? nodeRelativePortDistributor : layerTotalPortDistributor;

            // The fixed layer is randomized
            minimizeCrossings(fixedLayer, crossminHeuristic, forward, false, true);

            // Reset last and current run crossing counters
            int curSweepCrossings = Integer.MAX_VALUE;
            int prevSweepCrossings = Integer.MAX_VALUE;

            // Do alternating forward and backward sweeps as long as the number of crossings
            // decreases with respect to the previous sweep.
            boolean firstSweep = true;
            do {
                // The formerly current sweep is now the previous sweep
                copySweep(curSweep, prevSweep);
                prevSweepCrossings = curSweepCrossings;
                curSweepCrossings = 0;

                // count in-layer crossings
                curSweepCrossings += inlayerCrossingsCounter.countCrossings(fixedLayer, fixedLayerIndex);

                if (forward) {
                    // Perform a forward sweep
                    for (int layerIndex = 1; layerIndex < layerCount; layerIndex++) {
                        NodeGroup[] freeLayer = curSweep[layerIndex];

                        portDistributor.calculatePortRanks(fixedLayer, PortType.OUTPUT);
                        minimizeCrossings(freeLayer, crossminHeuristic, true, !firstSweep, false);

                        // in-layer crossings
                        curSweepCrossings +=
                                inlayerCrossingsCounter.countCrossings(freeLayer, layerIndex);
                        // between-layers crossings
                        if (hasHyperedgesWest[layerIndex] || hasHyperedgesEast[layerIndex - 1]) {
                            curSweepCrossings +=
                                    hyperedgeCrossingsCounter.countCrossings(fixedLayer, freeLayer);
                        } else {
                            curSweepCrossings +=
                                    normalCrossingsCounter.countCrossings(fixedLayer, freeLayer);
                        }

                        fixedLayer = freeLayer;
                    }
                    fixedLayerIndex = layerCount - 1;
                } else {
                    // Perform a backward sweep
                    for (int layerIndex = layerCount - 2; layerIndex >= 0; layerIndex--) {
                        NodeGroup[] freeLayer = curSweep[layerIndex];

                        portDistributor.calculatePortRanks(fixedLayer, PortType.INPUT);
                        minimizeCrossings(freeLayer, crossminHeuristic, false, !firstSweep, false);

                        // in-layer crossings
                        curSweepCrossings +=
                                inlayerCrossingsCounter.countCrossings(freeLayer, layerIndex);
                        // between-layers crossings
                        if (hasHyperedgesEast[layerIndex] || hasHyperedgesWest[layerIndex + 1]) {
                            curSweepCrossings +=
                                    hyperedgeCrossingsCounter.countCrossings(freeLayer, fixedLayer);
                        } else {
                            curSweepCrossings +=
                                    normalCrossingsCounter.countCrossings(freeLayer, fixedLayer);
                        }

                        fixedLayer = freeLayer;
                    }
                    fixedLayerIndex = 0;
                }

                // Switch the sweep direction
                firstSweep = false;
                forward = !forward;
            } while (curSweepCrossings < prevSweepCrossings && curSweepCrossings > 0);

            // Compare the current result with the best one
            if (curSweepCrossings < bestSweepCrossings || prevSweepCrossings < bestSweepCrossings) {
                // Restore the previous sweep's ordering if it has become worse
                if (curSweepCrossings <= prevSweepCrossings) {
                    copySweep(curSweep, bestSweep);
                    bestSweepCrossings = curSweepCrossings;
                } else {
                    copySweep(prevSweep, bestSweep);
                    bestSweepCrossings = prevSweepCrossings;
                }
            }
        }

        // Apply the ordering to the original layered graph
        ListIterator<Layer> layerIter = layeredGraph.getLayers().listIterator();
        while (layerIter.hasNext()) {
            Layer layer = layerIter.next();
            NodeGroup[] nodes = bestSweep[layerIter.previousIndex()];
            ListIterator<LNode> nodeIter = layer.getNodes().listIterator();
            while (nodeIter.hasNext()) {
                nodeIter.next();
                nodeIter.set(nodes[nodeIter.previousIndex()].getNode());
            }
        }

        // In the old days, the ports were distributed at this point in time. This has been moved to a
        // separate processor, the PortDistributionProcessor.

        dispose();
        monitor.done();
    }

    /**
     * Minimize crossings between the given layer and its preceding or subsequent layer.
     *
     * @param layer the layer that is to be reordered
     * @param heuristic the crossing minimization heuristic
     * @param forward if true the preceding layer is taken as fixed layer, otherwise the subsequent
     *          layer is taken
     * @param preOrdered whether the nodes of the given layer are already ordered
     * @param randomize whether to randomize all node positions
     */
    private void minimizeCrossings(final NodeGroup[] layer,
            final ICrossingMinimizationHeuristic heuristic,
            final boolean forward, final boolean preOrdered, final boolean randomize) {

        List<NodeGroup> nodeGroups = Lists.newArrayList();
        for (NodeGroup ng : layer) {
            nodeGroups.add(ng);
        }

        // minimize crossings in the given layer
        heuristic.minimizeCrossings(nodeGroups, preOrdered, randomize, forward);

        // apply the new ordering
        int index = 0;
        for (NodeGroup nodeGroup : nodeGroups) {
            for (LNode node : nodeGroup.getNodes()) {
                layer[index++] = node.getProperty(InternalProperties.NODE_GROUP);
            }
        }
    }

    // /////////////////////////////////////////////////////////////////////////////
    // Utility Methods

    /**
     * Copy the content of the source node array to the target node array.
     *
     * @param source
     *            a layered graph
     * @param dest
     *            a node array to copy the graph into
     */
    private static void copySweep(final NodeGroup[][] source, final NodeGroup[][] dest) {
        for (int i = 0; i < dest.length; i++) {
            for (int j = 0; j < dest[i].length; j++) {
                dest[i][j] = source[i][j];
            }
        }
    }

}

redpajama_set_name: RedPajamaGithub


text:
\section{Introduction.\label{sec:introduction}}
\input{introduction}

\section{Non-standard kinetic terms: a brief review.\label{sec:non-standard}}
\input{non-standard}

\section{Generalized $\beta$-function formalism.\label{sec:formalism}}
\input{formalism}

\section{Models with non-standard kinetic terms.\label{sec:general}}
\input{general}

\subsection{DBI-like models.\label{sec:DBI}}
\input{DBI}

\subsection{A new class of models.\label{sec:new_models}}
\input{new_models}

\section{Conclusions.\label{sec:conclusions}}
\input{conclusions}

\subsection*{ Acknowledgements}
M.P. would like to thank E. Kiritsis and F. Vernizzi for useful discussions.\\
P.B. and M.P. acknowledge the financial support of the UnivEarthS
Labex program at Sorbonne Paris Cit\'e (ANR-10-LABX-0023 and
ANR-11-IDEX-0005-02). J.M. is supported by Principal's
Career Development Scholarship and Edinburgh Global Research Scholarship. \\
J.M. and M.P. want to dedicate this paper to the memory of Pierre Bin\'etruy who passed away right after the end of the reviewing process. Pierre was not only a great physicist but also a great person and probably the best ``boss'' we could have dreamed of.

\newpage

redpajama_set_name: RedPajamaArXiv


text:
Khorasan Razavi Province Boosting Energy Projects
EghtesadOnline: The Energy Ministry has 46 power, water and wastewater projects in cities and villages of Khorasan Razavi Province, the minister said Monday.
"With $1.5 billion investment, the projects have made 51% progress and most are expected to come on line by 2022," Reza Ardakanian was quoted as saying by the ministry news portal Paven.
He made the remarks at the inauguration ceremony of Zakariya 400kv power substation in Mashhad.
Another 400kv substation was launched in the holy city Mashhad on Monday at a cost of $7.7 million, according to Financial Tribune.
Thirteen other projects were launched Monday in different counties of the northeastern province and cost $44 million.
Among the projects was the Abivard Dam in Dargaz County, costing $16.5 million, which will provide water to households, farms and industris in the region.
The diversion dam can hold 43 million cubic meters of water. It supplies water to 69,000 people in Dargaz, Lotfabad and Nokhandan cities plus rural areas to the south of the dam.
With the completion of a project to supply potable water to 30 villages in the province, 2,500 people in rural areas have now access to fresh water.
Another project was the wastewater network for the agro sector in Sabzevar County costing $1.7 million.
The Energy Ministry has plans to inaugurate 227 power and water projects across Iran in the second half of the current year (Sept 2019-March 2020) with a total investment of 2.5 billion.
Boost Projects Iran Khorasan Razavi energy projects Water Wastewater power
Water, Wastewater, Power Projects in Fars to Cost $586m
South Khorasan Strongly Pushing Water, Power, Wastewater Agenda
Major Water, Power Projects Underway in Fars Province
Rural Water Supply Remains Energy Ministry Priority
Power, Water Supply Capacity Increases in Western Regions


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Home/About INS/INS Committees/Student Liaison Committee
Student Liaison CommitteeDavis Schoenfeld2019-07-01T18:44:58+00:00
Welcome to INS-SLC
Thank you for visiting the page of the INS Student Liaison Committee! You may read about the history of SLC and about our student committee members on this page, or connect with the INS-SLC on their Facebook page.
Click here to connect with INS-SLC on Facebook
The INS Student Liaison Committee also operates a google group especially for students. The google group is intended to provide a novel means of communication amongst neuropsychology trainees on professional development topics. The google group is a place where students can pose questions regarding training, provide peer consultation, and share ideas. The SLC will also post INS programming updates and upcoming deadlines.
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History of the INS-SLC
At the 38th Annual Meeting of INS in Acapulco in 2010, the Board or Governors organized a meeting to solicit comments from students and trainees about how to better serve associate members and increase their involvement. As a result of this meeting, the Society formed a student-run standing committee with specific goals to address student and trainee professional development needs, foster student contributions to neuropsychological science, and provide opportunities for student involvement and leadership within the INS
Donations to the Student Activities Fund supports the Student Liaison Committee in fostering student leadership, creating student-oriented programs and events, and developing funds and services to support students in the field of neuropsychology.
INS-SLC Committee Members
Jillian Tessier
Jillian Tessier is a third-year doctoral student in the Clinical Psychology program at Drexel University, in Philadelphia, PA, under the mentorship of Dr. Maria Schultheis. She earned her Bachelor's degree in Psychology from University of California, Berkeley in 2012. Her senior honors thesis investigated the link between insomnia and working memory in individuals with bipolar disorder. She has worked as a research coordinator on several projects at the Department of Veterans Studies, including evaluating exercise's effect on memory in mild cognitive impairment, and driving behaviors and anxiety in PTSD. Presently, her primary research interests include the impact of mild traumatic brain injury and insomnia on multitasking in vocational settings. She is interested in identifying predictors of successful ability to return to work after traumatic brain injury.
Anouk Smits
Anouk Smits is a PhD student in the Graduate School of Psychology at the University of Amsterdam (UvA), The Netherlands. She is affiliated to the University Medical Center Utrecht (UMCU), where she is conducting her research. Anouk completed a Master's Degree in Neuroscience and in Neuropsychology, both at Utrecht University. Her main research interests are within the combination of neuropsychology, neurology, and neuroimaging. In her doctoral research, she is associating brain connectivity measures to visual and visuomotor outcome in stroke and brain cancer.
Taylor Greif, M.S.
Annual Meeting Programs Representative
Taylor is completing her 3rd year in the Clinical Psychology PhD program at Saint Louis University under the mentorship of Dr. David Kaufman. She completed her bachelor's degree at DePaul University in Chicago where she studied traumatic brain injury in Dr. Dorothy Kozlowski's lab. She most recently completed her master's thesis on the electrophysiological effects of mindfulness meditation in college students, and her current research interests pertain to predicting outcomes after deep brain stimulation surgery in patients with Parkinson's disease. Before moving to St. Louis, Taylor worked as a psychometrist at the University of Illinois at Chicago, and she is currently a clinical neuropsychology extern at Saint Louis University Hospital and Washington University School of Medicine.
Felicity Evans
Mid-Year Programs Representative
Felicity Evans is in her second year of the Doctor of Psychology (Clinical Neuropsychology) program at Monash University, in Melbourne, Australia. She completed a bachelor's degree in Science, followed by an honours degree in Psychology in 2015. For her honours thesis, she investigated emotion recognition in Parkinson's disease. The following year, she pursued her interest in neurodegenerative conditions by undertaking a research internship at the National Parkinson's Foundation Centre of Excellence at King's College Hospital, London. For her current doctoral research project, she is investigating the efficacy of compensatory memory strategies in the rehabilitation of stroke survivors. Felicity currently holds the role of July Meeting Representative, and she is responsible for organising the student program at the Prague (2018), Rio de Janeiro (2019) and Vienna (2020) INS meetings.
Mary Godfrey
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Mary Godfrey is a third year doctoral student in the clinical psychology program at Drexel University. Originally from St. Louis, Missouri, Mary received her Bachelor's degree in Psychology from Rhodes College in Memphis, Tennessee. After graduating, she worked as a Special Education teacher through Teach for America. Later, she worked as a research analyst at Vanderbilt University's Memory & Alzheimer's Center, where she assisted in researching the relationship between heart health and brain health of older adults in the prodromal phase of Alzheimer's disease. Her research is focused in pediatric neuropsychology and her research interests include the cognitive impairment associated with genetic disorders, specifically focusing on memory impairments associated with Down Syndrome.
Nella Korhonen
International Liaison Representative
Nella Korhonen is a doctoral student in clinical neuropsychology at the University of Helsinki, Finland. She received her master's degree from the University of Tampere, Finland in 2012. After graduating, she did a traineeship in the European Civil Protection and Humanitarian Aid Operations of the European Commission in Brussels. She has also lived and studied in Hong Kong and Italy. Nella has been an active member in the Finnish Psychological Association and chaired their working group for Young Psychologists in 2016. While working on her grant-funded dissertation, she also works part-time as a neuropsychologist at the phoniatric outpatient clinic in Helsinki University Hospital. Her research project investigates ADHD symptoms from childhood to adulthood in a perinatal risk cohort. In the future, she hopes to continue to combine her research interest with her clinical experience in developmental neuropsychology.
Kalliopi Megari, MSc, BA
Kalliopi is a PhD candidate at Aristotle University of Thessaloniki in Greece with thesis subject: Neuropsychological outcome and quality of life following surgical myocardial revascularization: The influence of cognitive reserve, mood and type-D personality. She holds undergraduate degrees in Nursing and Psychology, as well as a Master's degree in Psychology from Aristotle University and also attended further education from Department of Educational and Social Policy, University of Macedonia, in people with special needs and disabilities. Kalliopi has many years of research experience working with patients who have neurological, psychiatric, and cognitive disorders, as well as chronic disease patients suffering from cancer and cardiovascular diseases. Her research interests include neuropsychological outcome and quality of life in chronic diseases patients such as cardiac surgery patients and cancer patients and cognitive reserve of central nervous system's diseases. Kalliopi was a research fellow from 2009-2015 in Department of Cardiothoracic Surgery in AHEPA Hospital. She has published articles in a wide range of international peer reviewed medical journals and participated in many international conferences. Her work has earned her many prestigious awards and distinctions worldwide. Kalliopi was a Member of the Editorial Board of Health Psychology Research journal (2013-2015) and she is an associate Editor in Psychology & Psychological Research International Journal (2018). She is a reviewer of papers for scientific journals in the field of Neuropsychology, Medicine and Quality of life. Kalliopi has given many lectures to undergraduate and postgraduate students at School of Psychology at Aristotle University of Thessaloniki and also at University of Warsaw, Department of Neuropsychology, in Poland.
Kalliopi has also clinical experience, offering psychological counseling to prisoners in the Thessaloniki Diavata Prison and being the Head of the Centre of creative activities of children with disabilities, a position that requires effectiveness, knowledge and strong willingness as well as empathy and love for the people. She was interviewed for Quality of Life experiences in Health Care Magazine Issue 3, (2014) with the title, Kalliopi Megari: "Giving a Voice to the Chronically Ill", Meet the Expert Column.
Corey Bolton
Corey Bolton is a third-year doctoral student in Clinical Psychology at Wheaton College in suburban Chicago. Broadly, Corey is interested in researching atypical presentations of Alzheimer's disease. His dissertation project is focused on the neurocognitive characteristics of early-onset Alzheimer's disease in patients with autopsy confirmed AD pathology. Clinically, he has received training in neuropsychology at the University of Chicago Medical Center and the Cognitive Neurology and Alzheimer's Disease Center at Northwestern Medicine. Further, he works as a research assistant at Loyola University Medical Center. Corey is glad to be part of the Student Liaison Committee and looks forward to representing the INS student membership as Member-at-Large.
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text:
Q: View list of comments I've posted There have been times when I post a comment on something and I want to see what others have replied to it. It would be nice if someone replying to an answer/question you commented on triggered the mail flag to go orange or if you could see a list of comments you've posted so you can go back and check.

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You won't get alerts for just any activity in a post you've commented on, but you will get alerts when someone specifically attempts to alert you in a comment sequence. See this FAQ post for more details.

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text:
Facebook Groups Aid Uneasy Rogers Park Residents After Seemingly Random Killings
By Ash-har Quraishi	• Published October 12, 2018 • Updated on October 13, 2018 at 10:04 pm
Some residents say the initially increased police presence here in Rogers Park has since diminished. So, they are watching each others backs.
Nearly two weeks after the execution-style murders police say they have credible leads but no arrests. Some in the neighborhood are still gripped with terror.
"There's some of us that are going out that think people are kind of returning to normalcy but there's a lot of people in this community that are still scared that don't even want to go out of their house," resident Jason Saini said.
At the lake front bike path near Loyola Park, candles and flowers mark the spot where 24-year-old Eliyahu Moscowitz was killed. He was shot in the head at point-blank range around 10:20 p.m. Oct. 1.
Just 36 hours earlier 73-year-old Douglas Watts was shot and killed while walking his dogs in the 1400 block of West Sherwin Avenue.
Police say both murders were "committed with the same weapon, likely by the same individual" seen in surveillance images.
The killings galvanized residents like Saini create block groups on Facebook. A place for residents to get support, connect, and find walking partners.
"There were privacy concerns so I thought, lets shrink it to a block level," he said. "Lets get our local blocks together, get everybody to meet and so we can kind of coordinate those walks and everybody can look out for each other."
Saini says that since the murders people have become more vigilant. They are more aware and watching out for one another.
"Like a lot of murders unfortunately that happen in Chicago it may not get solved at all so, a lot of these support networks that we're putting together in the community might have to sustain us for quite some time," he said.
Saini says other block groups have since popped up. Hundreds are connected in the private groups. He says they will remain connected as long as people fill uneasy.
Chicagofacebook


WASHINGTON, D.C. (March 22, 2018) – The Open Space Institute (OSI) today joined together with fellow members of the Land and Water Conservation Fund (LWCF) Coalition, a partnership of leading environmental and open space advocates, in expressing appreciation for the increase in LWCF funding included in the Fiscal Year 2018 Omnibus Appropriations Act. The expanded funding represents a win for conservation. The bipartisan legislation increases spending for the LWCF to $425 million in FY18, a $25 million increase from previous-year levels. The bill contains additional funding for a wildlife conservation account previously derived from LWCF, bringing the effective total to $436 million in land conservation spending.
The funding agreement means that many critical on-the-ground conservation and recreational access projects will receive funding from America's most important conservation program this year. More importantly, the agreement represents a bipartisan rejection of the Trump Administration's proposal to eliminate funding for the LWCF.
Though pleased with the funding increase, Elliman expressed concern that the victory could be short-lived; since reauthorization of the LWCF was not included in the bill, the current authorization for LWCF is set to expire on September 30, putting the program at risk unless Congress acts to save it from extinction.
"I am confident that the coalition will, again, join forces and make compelling arguments that the LWCF and all the good it accomplishes for our nation needs to be saved for future generations," said Elliman.


text:
\section{Introduction}

Since the onset of the twenty-first century the ultracold matter experiments have increasingly directed towards the formation of molecules from laser-cooled atoms or direct cooling molecules from higher temperatures~\cite{CarrNJP09,QuemenerCR12,BohnS17}. These investigations have been motivated by both fundamental and practical dimensions with possible applications ranging from controlled chemical reactions~\cite{KremsPCCP08,TomzaPRL15,BalakrishnanJCP16} and precision measurements~\cite{DeMilleScience17,SafronovaRMP18} to quantum simulation~\cite{MicheliNatPhys06,BaranovCR12} and quantum computation~\cite{DeMillePRL02,Albert2019}. Following the successful experiments with ultracold molecules, increasingly more trapped molecular ions have been studied at low temperatures~\cite{WillitschIRPC12,TomzaRMP19}. Molecular ions can be either produced at higher temperatures and subsequently cooled down~\cite{TongPRL10,StaanumNP10} or formed from ultracold ion-atom mixtures via association~\cite{SullivanPCCP11,TomzaPRA15a} or from ulracold molecules via ionization~\cite{JyothiPRL16,SchmidPRL18}.

High controllability of ultracold molecules is enhanced even further in case of molecular ions, as they exhibit significantly longer-range interactions than neutral species, and can be manipulated and detected on a single particle level in ion traps~\cite{WinelandRMP13,TomzaRMP19}. This can allow for high precision spectroscopy~\cite{LohScience13,GermannNatPhys14,KajitaPRA14} and measurements of chemical reaction rates with particle densities of molecular ions much smaller and better controlled than with neutral molecules~\cite{PuriNC19,Dorfler2019}. Moreover, the charged products of chemical reactions can be trapped thus opening the way for measuring product-state distributions and state-to-state reaction rates~\cite{ChangScience13,PuriScience17,KilajNC18}. Sympathetically cooled molecular ions such N$_2^+$~\cite{HallPRL12} and OH$^-$~\cite{DeiglmayrPRA12} were already immersed into ultracold Rb atoms, while BaCl$^+$ molecular ions were sympathetically cooled down by collisions with ultracold Ca atoms~\cite{RellergertNature13,StoecklinNP16}.

Enormous successes of experiments with ultracold alkali and alkaline-earth atoms and ions result from their electronic structures favorable for laser-cooling. Therefore, first cold ion-atom mixtures unsurprisingly employed alkaline-earth ions and alkali or alkaline-earth atoms~\cite{MakarovPRA03,GrierPRL09,HallMP13a,ZipkesNature10,HazePRA13,HallPRL11,RaviNatCommun12,HartePRL12,SmithAPB14,MeirPRL16,JogerPRA17}. The radiative formation of RbCa$^+$~\cite{HallPRL11, HallMP13b}, RbBa$^+$~\cite{HallMP13a}, CaYb$^+$~\cite{RellergertPRL11} and CaBa$^+$~\cite{SullivanPCCP11} molecular ions was already observed in cold collisions between respective ions and atoms. Rb$_2^+$~\cite{JyothiPRL16} and Ca$_2^+$~\cite{SullivanPCCP11} molecular ions were formed by the photoionization of ultracold molecules. Both approaches may be used for the formation of the Sr$_2^+$ molecular ion studied in this paper.

Neutral alkaline-earth dimers have been spectroscopically studied in a number of experiments, including  Be$_2$~\cite{MerrittPCCP08,MerrittS09}, Mg$_2$~\cite{BalfourCJP70,McCaffreyJCP88,KnockelJCP13}, Ca$_2$~\cite{BalfourCJP75,vidalJCP80,bondybeyCPL84,AllardPRA02,AllardEPJD05}, Sr$_2$~\cite{BergemanJCP80,GerberJCP84,SteinPRA08,SteinEPJD10,SteinEPJD11}, Ba$_2$~\cite{boutouZP97,lebeaultJMS98}.
They also received considerable attention as the subject of theoretical studies~\cite{JonesJCP79,PartridgeJCP90,AlloucheCP95,CzuchajTCA03,BusseryPRA03,BusseryPRA05,LeeJPCA05,BusseryJCP06,BusseryMP06,PatkowskiJPCA07,PatkowskiS09,
BouissouJCP10,FromagerPRA10,heavenCPL11,LiJPCA11,SchaferPRA07,
yangTCA12,KerkinesJCP12,LiJPCA11,AmitayFD11,MeshkovJCP14,SharmaJCP14,WeiCPL15,KalemosJCP16,nasiriCTC17,StephenMP19,LesiukJCTC19}, including several works on Sr$_2$~\cite{BoutassettaPRA96,WangJPCA00,BeloyPRA11,czuchajCPL03,KotochigovaJCP08,KochPRA08b,SkomorowskiPRA12,SkomorowskiJCP12}.
Recently, ground breaking experiments with an ultracold gas of Sr$_2$ molecules in an optical lattice have been realized~\cite{ZelevinskyPRL06,ReinaudiPRL12,StellmerPRL12,CiameiPRA17} to study both high precision spectroscopy~\cite{ZelevinskyPRL08,McGuyerPRL13,McGuyerNP15,McGuyerPRL15,KondovNP19} and controlled photodissociation~\cite{mcdonaldN16,McDonaldPRL18,KondovPRL18}. Despite having only four valence electrons, the Sr$_2$ molecule turned out to be a challenging system for an accurate theoretical description of its electronic structure. An initial disagreement between relativistic calculations~\cite{KotochigovaJCP08} and experimental results~\cite{SteinPRA08} for excited electronic states was later resolved using higher level calculations~\cite{SkomorowskiJCP12}.

Surprisingly, there are only a few experimental or theoretical studies of alkaline-earth molecular ions. The dissociation energies were experimentally measured for Be$_2^+$~\cite{MerrittS09,AntonovJCP10} and Sr$_2^+$~\cite{dugourdCPL92} using photoionization of neutral dimers. The ground electronic state was theoretically investigated for all alkaline-earth molecular ions~\cite{ABC+,LiMP13}. The lowest excited states were studied for Be$_2^+$~\cite{BanerjeeCPL10}, Mg$_2^+$~\cite{StevensJCP77,AlharzaliJPB18}, and Ca$_2^+$~\cite{LiuPRA78,BanerjeeCPL12}. To the best of our knowledge, excited states of the Sr$_2^+$ molecular ion have not yet been investigated theoretically or experimentally.

Here, we fill this gap and calculate the electronic structure of the Sr$_2^+$ molecular ion using \textit{ab initio} methods of quantum chemistry. We characterize and benchmark two lowest electronic states of the Sr$_2^+$ molecular ion with a range of computational techniques including hierarchy of configuration interaction and coupled cluster methods. Next, we use multireference configuration interaction method restricted to single and double exctiations to obtain potential energy curves for 41 excited electronic states and electric dipole transition moments between them. 11 excited states are also obtained with the coupled cluster method restricted to single, double, and noniterative triple excitations. We show that the Sr$_2^+$ molecular ion despite of its apparently simple structure with three valence electrons only is challenging for state-of-the-art quantum chemistry methods due to its multireference nature and high density of states. Finally, we provide Franck-Condon factors governing the photoionization of ground-state Sr$_2$ molecules into $^2\Sigma^+_u$ and $^2\Sigma^+_g$ states of Sr$_2^+$ molecular ions.  The Sr$_2^+$ molecular ion is an interesting system for theoretical studies because it can be produced and investigated in modern experiments with ultracold Sr$_2$ molecules or with mixtures of Sr$^+$ ions immersed into ultracold Sr atoms. Thus, our results may be useful for guiding future formation and spectroscopic measurements of the Sr$_2^+$ molecular ion in the ground and excited electronic states.

The paper has the following structure. Section~\ref{sec:theory} concerns theoretical methods used in the \textit{ab initio} electronic structure calculations. Section~\ref{sec:results} presents and discusses the results obtained for the ground and excited electronic states of the Sr$_2^+$ molecular ion. Section~\ref{sec:summary} summarizes the paper and points to applications and extensions of the presented results.

\section{Computational details}
\label{sec:theory}

The Sr$_2^+$ molecular ion is composed of a closed-shell strontium atom with two valence electrons interacting with an open-shell strontium ion with one valence electron. Therefore, the resulting system has three valence electrons with a doublet multiplicity of the ground electronic state, while excited states can be either doublets or quartets. Excited electronic states correlate with atomic thresholds resulting from exciting the Sr$^+$ion or Sr atom, or both of them (see Tables~\ref{tab:atomic} and \ref{tab:thresholds}).

Here, to calculate potential energy curves in the Born-Oppenheimer approximation we adopt the computational scheme successfully applied to the ground and excited electronic states of the SrYb and Sr$_2$ molecules~\cite{TomzaPCCP11,SkomorowskiJCP12}, and LiYb$^+$ molecular ion~\cite{TomzaPRA15a}. All interaction and excitation energies, as well as electric dipole transition moments, are obtained with the multireference configuration interaction method restricted to single and double excitations (MRCISD) using orbitals optimized with the multi-configurational self-consistent field method (MCSCF)~\cite{MRCC} with a large active space composed of all molecular orbitals created from the $5s$, $5p$, $5d$, and $6s$ orbitals of both the Sr atom and the Sr$^+$ ion. The ground state and the lowest energetic excited electronic states in each irreducible representation of the $D_{2h}$ point group are additionally computed with the spin-restricted open-shell coupled cluster method restricted to single, double, and non-iterative triple excitations (RCCSD(T))~\cite{PurvisJCP82,KnowlesJCP93}. For comparison, we also present results for the $X^2\Sigma_u^+$ ground and the $^2\Sigma_g^+$ lowest-energetic excited state with the following methods: open-shell M$\o$ller-Plesset theory (RMP2)~\cite{MP2}, the spin-restricted open-shell coupled cluster method restricted to single and double excitations (RCCSD)~\cite{KnowlesJCP93}, MRCISD with the Davidson correction (MRCISD+Q)~\cite{MRCC}, and configuration interaction method restricted to single and double excitations (CISD)~\cite{MRCC}.
The interaction energy,~$E_\text{int}$, is computed as the difference between the energy of the dimer in a given state,~$E_{\text{Sr}_2^+(^{2S+1}\Lambda)}$, and the energies of the atom,~$E_{\text{Sr}(^{2s+1}L)}$, and ion,~$E_{\text{Sr}^+(^{2s'+1}L')}$, in the electronic states corresponding to the dissociation limit of the dimer in the~$^{2S+1}\Lambda$~state,
\begin{equation}
E_\text{int}=E_{\text{Sr}_2^+(^{2S+1}\Lambda)}-E_{Sr(^{2s+1}L)}-E_{Sr^+(^{2s'+1}L')}\,.
\end{equation}
All results are obtained by applying the Boys-Bernardi counterpoise correction method~\cite{BoysMP70}. All electronic structure calculations are performed with the \textsc{Molpro} package of \textit{ab initio} programs \cite{Molpro,MOLPRO-WIREs}.

\begin{table}[tb]
\caption{Atomic excitation energies of the Sr~atom and the Sr$^+$~ion, and the ionization potential of the Sr~atom calculated with the MRCISD and CCSD(T) methods compared with experimental data. All energies are in cm$^{-1}$.\label{tab:atomic}}
\begin{ruledtabular}
\begin{tabular}{lrrr}
Electronic transition    & MRCI & CC & Exp.~\cite{NIST} \\
\colrule
Sr$(^1S$ $\to$ $^3P)$   &  14139      &    14641    & 14703 \\
Sr$(^1S$ $\to$ $^3D)$   &  19071  &  18745 &  18254 \\
Sr$^+(^2S$ $\to$ $^2D)$   &  15553  &   15371 & 14743  \\
Sr$^+(^2S$ $\to$ $^2D)$   & 23839  & 24152 & 24249   \\
Sr$(^1S)$ $\to$ Sr$^+(^2S)$ & 44824 & 45820 & 45932 \\
\end{tabular}
\end{ruledtabular}
\end{table}

Scalar relativistic effects are included by replacing 28~inner-shell electrons in both the Sr atom and the Sr$^+$ ion with the small-core, fully relativistic, energy-consistent pseudopotential ECP28MDF~\cite{LimJCP05} from the Stuttgart  library. Thus, in the present study the Sr$_2^+$ molecular ion is treated as a system of effectively correlated 19~electrons. The additional advantage of employing pseudopotentials is the possibility to use larger basis sets to describe the valence electrons, while the inner-shell electron density is reproduced with the accuracy of high-quality atomic calculations. To this end, we utilize the large [14s11p6d5f4g] basis set proposed in Refs.~\cite{TomzaPCCP11,SkomorowskiJCP12}, augmented by the set of the [3s3p2d2f1g] bond functions~\cite{midbond}. The electric dipole transition moments are computed using respective MRCISD wave functions.

\begin{table*}[tb!]
\caption{Asymptotic energies (in cm$^{-1}$) and molecular states arising from different states of the strontium atom and ion. MRCI excitation energies  are averaged out over separate calculations for different molecular symmetries due to the size inconsistency of the MRCI method. \label{tab:thresholds}}
\begin{ruledtabular}
\begin{tabular}{lrrrl}
Asymptote    & MRCI energy & CC energy & Exp.~energy~\cite{NIST} &  Molecular states \\
\colrule
Sr$^+(^2S)$~+~Sr$(^1S)$   &  0      &    0    & 0  & $^2\Sigma_g$, $^2\Sigma_u^+$  \\
Sr$^+(^2S)$~+~Sr$(^3P)$   &  14100  &  14612 &  14703 &  $^2\Sigma_g$,$^2\Sigma_u^+$, $^2\Pi_g$,  $^2\Pi_u$, $^4\Sigma_g^+$, $^4\Sigma_u^+$, \\
& & & & $^4\Pi_g$, $^4\Pi_u$ \\
Sr$^+(^2D)$~+~Sr$(^1S)$   &  16850  &   15843 & 14743 & $^2\Sigma_g^+$, $^2\Sigma_u^+$, $^2\Pi_g$, $^2\Pi_u$, $^2\Delta_g$, $^2\Delta_u$,    \\
Sr$^+(^2S)$~+~Sr$(^3D)$   & 19100  & 18770 & 18254 &  $^2\Sigma_g$, $^2\Sigma_u^+$, $^2\Pi_g$, $^2\Pi_u$, $^2\Delta_g$, $^2\Delta_u$,    \\
                    &        &       &         & $^4\Sigma_g^+$, $^4\Sigma_u$, $^4\Pi_g$, $^4\Pi_u$, $^4\Delta_g$, $^4\Delta_u$ \\
Sr$^+(^2S)$~+~Sr$(^1D)$   & 20950  & - & 20150  &$^2\Sigma_g^+$, $^2\Sigma_u$, $^2\Pi_g$, $^2\Pi_u$, $^2\Delta_g$, $^2\Delta_u$,    \\
Sr$^+(^2S)$~+~Sr$(^1P)$   & 23000  & - &  21698 & $^2\Sigma_g^+$, $^2\Sigma_u$, $^2\Pi_g$, $^2\Pi_u$    \\
Sr$^+(^2P)$~+~Sr$(^1S)$   & 24100 & - & 24249 &$^2\Sigma_g^+$, $^2\Sigma_u$, $^2\Pi_g$, $^2\Pi_u$    \\
\end{tabular}
\end{ruledtabular}
\end{table*}

To evaluate the ability of the employed \textit{ab initio} approaches to reproduce experimental results, we first compared the theoretical excitation energies of a strontium atom and a strontium ion obtained in this study to the experimental values~\cite{NIST} (see Table~\ref{tab:atomic}). The CCSD(T) excitation energies of Sr and Sr$^+$ to the lowest $P$~state agree with the experimental results within 0.5\%, while the excitations to the lowest $D$~state agree within 3\% for the neutral atom and within 5\% for the ion. Additionally, the CCSD(T) ionization energy of the ground-state Sr atom to the ground-state ion is within 0.25\% from the experimental value. The MRCISD method yields slightly less accurate results: the errors of the excitation energies for the neutral strontium atom amount to 3.8\% for the $P$~state and to 4.5\% for the $D$~state. As for the strontium ion, the errors are equal to 1.7\% for the $P$ state and to 5.6\% for the $D$~state excitation. Additionally, the MRCISD ionization energy is within 2.4\% from the experimental value. Therefore, for both species, the excitations into the $D$ states are less accurately described by the basis set we employ than the $P$~ones, with the Sr$^+$ ion $D$~state exhibiting the lowest accuracy. However, further basis extension with additional $d$~orbitals does not improve the accuracy meaningfully. The static electric dipole polarizabilities of the Sr atom and Sr$^+$ ion obtained with the CCSD(T) method are 199.2$\,$a.u.~and 92.0$\,$a.u. and agree within 1\% with experimental values of 197.1(2)$\,$a.u.~and 91.3(9)$\,$a.u.~\cite{MitroyJPB10}, respectively.

\section{Results}
\label{sec:results}

\subsection{Potential energy curves}

We have calculated potential energy curves~(PECs) and their spectroscopic characteristics for the doublet and quartet gerade and ungerade electronic states of the Sr$_2^+$ molecular ion corresponding to the first seven lowest dissociation limits, Sr$^+$($^2$S)+Sr($^1S$), Sr$^+$($^1S$)+Sr($^3P$), Sr$^+$($^2D$)+Sr($^1S$), Sr$^+$($^2S$)+Sr($^3D$), Sr$^+$($^2S$)+Sr($^1D$), Sr$^+$($^2S$)+Sr($^1P$), and Sr$^+$($^2P$)+Sr($^1S$), that gives in total 42~electronic states. Molecular electronic states arising from atomic asymptotes are collected in Table~\ref{tab:thresholds}.

The $X^2\Sigma_u^+$ ground state potential energy curve is calculated by a range of \textit{ab initio} methods and presented in Fig.~\ref{fig:X}, while corresponding spectroscopic parameters are collected in Table~\ref{tab:su}. In the naive molecular orbital theory picture, the Sr$_2^+$ molecular ion in the $X^2\Sigma^+_u$ ground state can be considered as a bound molecule because its valence molecular configuration, $\sigma_g^2\sigma_u^{*1}$, has one more bonding than antibonding electron resulting in a bond of order one-half. Therefore, all employed electronic structure methods describe the ground-state Sr$_2^+$ molecular ion qualitatively correctly, albeit to a varying degree of accuracy.  Compared to the experimental value of dissociation energy at $8800(130)$~cm$^{-1}$~\cite{dugourdCPL92}, the MRCISD method with the Davidson correction returns the most accurate result, followed closely by CCSDT~\cite{MRCC}, which includes a full triple excitations in the CC scheme, and the regular MRCISD and CCSD(T) methods. Other approaches are less accurate: the CCSD method exhibits an error larger than 5\%, both MP2 and CISD methods
give as much as 15\% error, while for restricted Hartree-Fock method the error increases further to 30\% .

\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.98\columnwidth]{fig1}
\end{center}
\caption{Non-relativistic potential energy curves for (a)~the $X^2\Sigma^+_u$ ground and (b)~the $1^2\Sigma^+_g$ lowest-energetic excited electronic states of the Sr$_2^+$ molecular ion calculated at different levels of theory.}
\label{fig:X}
\end{figure}

In contrast to the ground state of the Sr$_2^+$ molecular ion, the 1$^2\Sigma_g^+$ lowest-energetic excited electronic state in the naive molecular orbital theory picture has one more antibonding than bonding electrons in its valence molecular configuration, $\sigma_g^1\sigma_u^{*2}$. For this reason, it is a challenging system for several \textit{ab initio} methods, which we compare in Fig.~\ref{fig:X} and Table~\ref{tab:su}. For instance, it is interesting to note that the M$\o$ller-Plesset interaction energy curve lies above the restricted Hartree-Fock one for intermediate distances from around 15 to 30 bohrs. This behavior can be justified by the fact that both monomers (Sr and Sr$^+$) are single-reference species, and therefore are well described already on the MP2 level, while the dimer has a multireference character which can be seen in the large differences between single and multireference correlated methods. For the same reason, coupled cluster methods restricted to single and double excitations fail to describe the system with size consistency, while it is preserved in the RCCSD(T) and RCCSDT methods. Note parenthetically that the curves in Fig.~\ref{fig:X} are shifted to the same asymptotic limit for a better visibility.

\begin{table}[tb]
\caption{Spectroscopic characteristics of the Sr$_2^+$ molecular ion in the $X^2\Sigma^+_u$ ground and $1^2\Sigma^+_g$ lowest-excited electronic states: equilibrium bond length~$R_e$, well depth~$D_e$, harmonic constant~$\omega_e$, and rotational constant~$B_e$, calculated at different levels of theory. \label{tab:su}}
\begin{ruledtabular}
\begin{tabular}{lrrrr}
Ref. & $R_e\,$(bohr) & $D_e\,$(cm$^{-1}$) &  $\omega_e\,$(cm$^{-1}$) & $B_e\,$(cm$^{-1}$) \\
\hline
\multicolumn{5}{c}{$X^2\Sigma^+_u$} \\
RHF & 8.33 & 6206 & 71.2 & 0.0198 \\
MP2 & 7.88 & 7569 & 80.1 & 0.0221  \\
CISD  & 8.00 & 7461 & 79.2 & 0.0215 \\
CCSD & 7.96 & 8245 & 79.5 & 0.0217  \\
CCSD(T)~\cite{ABC+} & 7.93 & 8576 &  79.9 & 0.0219 \\
CCSDT &  7.93 & 8703 & 80.1 & 0.0219 \\
MRCISD  & 7.99 & 8611 & 78.9 & 0.0215  \\
MRCISD+Q & 7.94 & 8801 & 80.2 & 0.0218 \\
Theo.~\cite{LiMP13} & 7.90 & 8745 & 80.7 & 0.022 \\
Exp.~\cite{dugourdCPL92} & - & 8800(130) & 86(3) & - \\
\hline
\multicolumn{5}{c}{$1^2\Sigma^+_g$} \\
RHF &  20.46 & 30 & 3.3 & 0.00328\\
MP2 &  \multicolumn{4}{c}{repulsive} \\
CISD  &  19.29 & 56 & 4.8 & 0.00369 \\
CCSD & 18.33 & 108 & 5.1 & 0.00409  \\
CCSD(T) & 10.03 & 78 & 23.7 & 0.01365 \\
2$^\text{nd}$ min. & 17.39 & 126 & 5.0 & 0.00455 \\
CCSDT & 9.55 & 359 & 29.5 & 0.01506 \\
2$^\text{nd}$ min. &  17.06 & 106 & 5.0 & 0.00472 \\
MRCISD  &  9.23 & 678 & 37.5 & 0.01612 \\
2$^\text{nd}$ min.  &  17.04 & 184 & 6.3 & 0.00473 \\
MRCISD+Q &  9.24 & 506 & 34.7 & 0.01608 \\
2$^\text{nd}$ min.  & 16.90 & 153 & 5.9 & 0.00481  \\
\end{tabular}
\end{ruledtabular}
\end{table}

The next significant feature of the 1$^2\Sigma_g^+$ curve is the presence of two minima: the short-range and the long-range ones. Similar double well structure for this electronic state was also theoretically predicted for the Be$_2^+$~\cite{BanerjeeCPL10} and Ca$_2^+$~\cite{BanerjeeCPL12} dimers. Less accurate and single-reference methods, like RHF, CISD and CCSD, detect only the latter minimum. The interactions at distances around the long-range minimum should be dominated by the induction component, however a more detailed analysis of this issue with e.g.~symmetry adapted perturbation theory~\cite{SAPT} is challenging due to the Sr$_2^+$ homonuclear structure and charge delocalization. The addition of non-iterative or full triples excitations in, respectively, the CCSD(T) and CCSDT methods results in the second minimum formation, which is further deepened by the MRCISD and MRCISD+Q methods. Thus, the existence of the short-range minimum arises from the multireference nature of the 1$^2\Sigma_g^+$ state at short distances in agreement with previous predictions for other alkaline-earth diatomic ions~\cite{BanerjeeCPL10,BanerjeeCPL12}. The analysis of the highest excitation amplitudes in the CCSD versus MRCISD methods shows that the two most important configuration functions (apart from the reference one) are strongly underestimated within the former approach. For instance, the double excitation from the highest occupied $\sigma^*_u$ orbital into the singly occupied $\sigma_g$ and the lowest unoccupied $\sigma_g$ is present in CCSD with the coefficient of about 0.3 for distance 8.5 bohr, while for MRCISD its contribution amounts to 0.8 (after applying intermediate normalization for MRCI). The situation is analogous for the second most important configuration which describes the excitation of both electrons from the same occupied $\sigma^*_u$ orbital into the lowest unoccupied $\sigma_g$ orbital (the CCSD coefficient of 0.2 is about two times too small when compared to the MRCISD one). These serious deficiencies of the CCSD model can explain the lack
\begin{figure}[hptb!]
\begin{center}
\includegraphics[width=0.96\columnwidth]{fig2}
\end{center}
\caption{Non-relativistic potentials energy curves for the $^2\Sigma^+_g$ and $^4\Sigma^+_g$ electronic states of the Sr$_2^+$ molecular ion.}
\label{fig:sg}
\begin{center}
\includegraphics[width=0.96\columnwidth]{fig3}
\end{center}
\caption{Non-relativistic potentials energy curves for the $^2\Sigma^+_u$ and $^4\Sigma^+_u$ electronic states of the Sr$_2^+$ molecular ion.}
\label{fig:su}
\end{figure}
\begin{figure}[hptb!]
\begin{center}
\includegraphics[width=0.96\columnwidth]{fig4}
\end{center}
\caption{Non-relativistic potentials energy curves for the $^2\Pi_g$ and $^4\Pi_g$ electronic states of the Sr$_2^+$ molecular ion.}
\label{fig:pg}
\begin{center}
\includegraphics[width=0.96\columnwidth]{fig5}
\end{center}
\caption{Non-relativistic potentials energy curves for the $^2\Pi_u$ and $^4\Pi_u$ electronic states of the Sr$_2^+$ molecular ion.}
\label{fig:pu}
\end{figure}
\!\!of the short-range minimum, which can be reproduced by adding at least triples excitations.

\begin{table}[tb]
\caption{Spectroscopic characteristics of the Sr$_2^+$ molecular ion in $^2|\Lambda|_g$ electronic states: equilibrium bond length~$R_e$, well depth~$D_e$, harmonic constant~$\omega_e$, and rotational constant~$B_e$ obtained with the MRCISD and RCCSD(T) methods. \label{tab:dubletyg}}
\begin{ruledtabular}
\begin{tabular}{lcrrrr}
State & Ref. & $R_e\,$(bohr) & $D_e\,$(cm$^{-1}$) &  $\omega_e\,$(cm$^{-1}$) & $B_e\,$(cm$^{-1}$) \\
\hline
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^1S)$} \\
$1^2\Sigma^+_g$  & MRCI &  9.23 & 678 & 37.5 & 0.0161 \\
$1^2\Sigma^+_g$  & CC & 10.03 & 78 & 23.7 & 0.0137 \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^3P)$} \\
$2^2\Sigma^+_g$  & MRCI  & 7.45 & 8027 & 104.4 & 0.0247  \\
$1^2\Pi_g$  & MRCI &  7.78 & 6044 & 105.2 & 0.0227 \\
$1^2\Pi_g$  & CC & 7.75 & 6041 & 81.5 & 0.0229  \\
\multicolumn{6}{c}{Sr$^+(^2D)$~+~Sr$(^1S)$} \\
$3^2\Sigma^+_g$  & MRCI & 8.46 & 5630 & 64.8 & 0.0192   \\
$2^2\Pi_g$  & MRCI & 7.77 & 7764 & 80.8 & 0.0228  \\
$1^2\Delta_g$  & MRCI & 7.20 & 10710 & 100.7 & 0.0265 \\
$1^2\Delta_g$  & CC & 7.23 & 9256 & 98.7 & 0.0241 \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^3D)$} \\
$4^2\Sigma^+_g$  & MRCI & 6.54 & 3013 & 422.7 & 0.0321  \\
$3^2\Pi_g$  & MRCI & 9.20 & 2182 & 48.0 & 0.0163 \\
$2^2\Delta_g$  & MRCI & 9.04 & 2425 & 48.7 & 0.0168 \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^1D)$} \\
$5^2\Sigma^+_g$  & MRCI &  10.60 & 1630 & 31.4 & 0.0122 \\
$4^2\Pi_g$  & MRCI & 8.36 & 2260 & 57.0 & 0.0197  \\
$3^2\Delta_g$  & MRCI & 13.51 & 702 & 33.6 & 0.0075 \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^1P)$} \\
$6^2\Sigma^+_g$  & MRCI  & 10.72 & 37 & 21.9 & 0.0120  \\
$5^2\Pi_g$  & MRCI &  12.21 & 1313 & 49.1 & 0.0092 \\
\multicolumn{6}{c}{Sr$^+(^2P)$~+~Sr$(^1S)$} \\
$7^2\Sigma^+_g$  & MRCI  & 14.70 & 891 & 15.6 & 0.0064  \\
$6^2\Pi_g$  & MRCI & 13.25 & 786 & 23.0 & 0.0078  \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}[tb]
\caption{Spectroscopic characteristics of the Sr$_2^+$ molecular ion in $^2|\Lambda|_u$ electronic states: equilibrium bond length~$R_e$, well depth~$D_e$, harmonic constant~$\omega_e$, and rotational constant~$B_e$ obtained with the MRCISD and RCCSD(T) methods. \label{tab:dubletyu}}
\begin{ruledtabular}
\begin{tabular}{lcrrrr}
State & Ref. & $R_e\,$(bohr) & $D_e\,$(cm$^{-1}$) &  $\omega_e\,$(cm$^{-1}$) & $B_e\,$(cm$^{-1}$) \\
\hline
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^1S)$} \\
$X^2\Sigma^+_u$  & MRCI & 7.99 & 8611 & 78.9 & 0.0215 \\
$X^2\Sigma^+_u$  & CC & 7.93 & 8576 &  79.9 & 0.0219 \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^3P)$} \\
$2^2\Sigma^+_u$  & MRCI  & 7.99 & 5345 & 79.3 & 0.0215  \\
$1^2\Pi_u$  & MRCI &  6.78 & 15290 & 93.2 & 0.0299 \\
$1^2\Pi_u$  & CC &  6.70 & 17030 & 125.9 & 0.0306 \\
\multicolumn{6}{c}{Sr$^+(^2D)$~+~Sr$(^1S)$} \\
$3^2\Sigma^+_u$  & MRCI & 12.22 & 2742 & 124.7 & 0.0092   \\
$2^2\Pi_u$  & MRCI & 7.59 & 8908 & 83.6 & 0.0239  \\
$1^2\Delta_u$  & MRCI & 8.10 & 7939 & 73.7 & 0.0210 \\
$1^2\Delta_u$  & CC &  7.89 & 6466 & 76.3 & 0.0215 \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^3D)$} \\
$4^2\Sigma^+_u$  & MRCI & 10.12 & 1876 & 62.3 & 0.0134  \\
$3^2\Pi_u$  & MRCI &  9.36 & 1761 & 44.4 & 0.0157 \\
$2^2\Delta_u$  & MRCI &  8.63 & 3575 & 52.2 & 0.0185\\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^1D)$} \\
$5^2\Sigma^+_u$  & MRCI &  8.82 & 2373 & 129.1 & 0.0178 \\
$4^2\Pi_u$  & MRCI & 11.88 & 1179 & 23.2 & 0.0097  \\
$3^2\Delta_u$  & MRCI & 13.88 & 656 & 30.0 & 0.0071 \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^1P)$} \\
$6^2\Sigma^+_u$  & MRCI  & 14.71 & 945 & 26.6 & 0.0064  \\
$5^2\Pi_u$  & MRCI &  7.49 & 1900 & 225.0 & 0.0245 \\
\multicolumn{6}{c}{Sr$^+(^2P)$~+~Sr$(^1S)$} \\
$7^2\Sigma^+_u$  & MRCI  & 13.58 & 378 & 15.6 & 0.0075 \\
$6^2\Pi_u$  & MRCI & 23.50 & 28 & 2.2 & 0.0025  \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}[tb]
\caption{Spectroscopic characteristics of the Sr$_2^+$ molecular ion in $^4|\Lambda|_g$ electronic states: equilibrium bond length~$R_e$, well depth~$D_e$, harmonic constant~$\omega_e$, and rotational constant~$B_e$ obtained with the MRCISD and RCCSD(T) methods. \label{tab:kwartetyg}}
\begin{ruledtabular}
\begin{tabular}{lcrrrr}
State & Ref. & $R_e\,$(bohr) & $D_e\,$(cm$^{-1}$) &  $\omega_e\,$(cm$^{-1}$) & $B_e\,$(cm$^{-1}$) \\
\hline
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^3P)$} \\
$1^4\Sigma^+_g$  & MRCI & 22.59 & 315 & 6.52 & 0.0027\\
$1^4\Sigma^+_g$  & CC &  \multicolumn{4}{c}{repulsive}  \\
$1^4\Pi_g$  & MRCI & \multicolumn{4}{c}{repulsive}   \\
$1^4\Pi_g$  & CC & \multicolumn{4}{c}{repulsive}  \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^3D)$} \\
$2^4\Sigma^+_g$  & MRCI  & 13.80 & 116 & 20.6 & 0.0072  \\
$2^4\Pi_g$  & MRCI &  8.68 & 2105 & 81.4 & 0.0182 \\
$1^4\Delta_g$  & MRCI  & 6.28 & 10857 & 145.2 & 0.0348  \\
$1^4\Delta_g$  & CC & 6.26 & 10529 & 140.5 & 0.0327  \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}[tb]
\caption{Spectroscopic characteristics of the Sr$_2^+$ molecular ion in $^4|\Lambda|_u$ electronic states: equilibrium bond length~$R_e$, well depth~$D_e$, harmonic constant~$\omega_e$, and rotational constant~$B_e$ obtained with the MRCISD and RCCSD(T) methods. \label{tab:kwartetyu}}
\begin{ruledtabular}
\begin{tabular}{lcrrrr}
State & Ref. & $R_e\,$(bohr) & $D_e\,$(cm$^{-1}$) &  $\omega_e\,$(cm$^{-1}$) & $B_e\,$(cm$^{-1}$) \\
\hline
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^3P)$} \\
$1^4\Sigma^+_u$  & MRCI & 9.29 & 9194 & 54.9 & 0.0159\\
$1^4\Sigma^+_u$  & CC & 9.20 & 9490 & 49.9 & 0.0162  \\
$1^4\Pi_u$  & MRCI & 7.63 & 8524 & 77.0 & 0.0236  \\
$1^4\Pi_u$  & CC & 7.56 & 9004 & 79.7 & 0.0241 \\
\multicolumn{6}{c}{Sr$^+(^2S)$~+~Sr$(^3D)$} \\
$2^4\Sigma^+_u$  & MRCI  & 8.91 & 3962 & 53.4 & 0.0173  \\
$2^4\Pi_u$  & MRCI & 9.27 & 2985 & 51.2 & 0.0160 \\
$1^4\Delta_u$  & MRCI  & 8.67 & 4161 & 51.4 & 0.0183  \\
$1^4\Delta_u$  & CC & 8.64 & 3229 & 52.9 & 0.0184  \\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.96\columnwidth]{fig6}
\end{center}
\caption{Non-relativistic potentials energy curves for the $^2\Delta_g$, $^4\Delta_g$, $^2\Delta_u$, and $^4\Delta_u$ electronic states of the Sr$_2^+$ molecular ion.}
\label{fig:dgu}
\end{figure}

Potential energy curves of the ground and excited doublet and quartet electronic states of $\Sigma_g^+$, $\Sigma_u^+$, $\Pi_g$, $\Pi_u$, $\Delta_g$, and $\Delta_u$ symmetries are presented in Figs.~\ref{fig:sg}-\ref{fig:dgu}. Their spectroscopic characteristics including equilibrium bond lengths~$R_e$, well depths~$D_e$, harmonic constants~$\omega_e$, and rotational constants~$B_e$ are gathered in Tables~\ref{tab:dubletyg}-\ref{tab:kwartetyu}. Based on the MRCISD results, we can identify that the $1^2\Pi_u$ state is the most strongly bound with $D_e=15290\,$cm$^{-1}$, followed by $1^4\Delta_g$ and $1^2\Delta_g$ with dissociation energies of 10857$\,$cm$^{-1}$ and 10710$\,$cm$^{-1}$, respectively. These three electronic states also exhibit the shortest equilibrium bond lengths of 6.78$\,$bohr, 6.28$\,$bohr, and 7.20$\,$bohr, respectively. On the other hand, the most weakly bound states are $6^2\Pi_u$, $6^2\Sigma_g^+$, and $2^4\Sigma_g^+$, for which the calculated dissociation energies are equal to 28$\,$cm$^{-1}$, 37$\,$cm$^{-1}$, and 116$\,$cm$^{-1}$, respectively. The largest equilibrium bond length is predicted for the $6^2\Pi_u$ state at 23.5$\,$bohr, followed by 22.6~bohr for the $2^4\Sigma_g^+$ state and 14.7$\,$bohr for the $6^2\Sigma_u^+$ state. We expect relative errors for deeply-bound electronic states to be significantly smaller than for weakly-bound ones.

By analyzing the pattern of the potential energy curves, we find that many of them display avoided crossings, suggesting strong radial non-adiabatic couplings between involved electronic states. We further observe a general tendency that lower-lying potential energy curves show a smooth behavior with well defined minima, while higher-lying states display perturbations, mostly in the form of avoided crossings due to the interaction with other electronic states of the same symmetry that result from closely-lying atomic thresholds. At high energies the density of electronic states becomes prohibitively large for MRCISD calculations, and for this reason we restricted  our analysis to the seven lowest-lying dissociation limits, although the highest-lying calculated states show avoided crossing with yet higher asymptotes.

\begin{figure*}[tb]
\begin{center}
\includegraphics[width=0.95\textwidth]{fig7}
\end{center}
\caption{Electric dipole transition moments between (a) the $X^2\Sigma^+_u$ ground and $^2\Sigma^+_g$ exited states, (b) the $X^2\Sigma^+_u$ ground and $^2\Pi_g$ exited states, (c) the $1^2\Sigma^+_g$ lowest excited and $^2\Sigma^+_u$ exited states, and (d) the $1^2\Sigma^+_g$ lowest excited and $^2\Sigma^+_u$ exited states of the Sr$_2^+$ molecular ion.}
\label{fig:dip}
\end{figure*}

The accuracy of our molecular calculations could be estimated by a comparison with the available experimental results. In the present case, we compare the obtained dissociation energy for the ground state of the Sr$_2^+$ ion with the experimental value of $8800(130)\,$cm$^{-1}$~\cite{dugourdCPL92}. The MRCISD value of 8611$\,$cm$^{−1}$ and the RCCSD(T) one of 8576$\,$cm$^{−1}$ correspond to the errors of 2.2\% and 2.6\%, respectively. The MRCISD+Q result agrees perfectly with the experimental value but the Davidson correction may give unreasonable results for excited states in the vicinity of avoided crossings. Unfortunately, no experimental excitation energies or dissociation energies for exited states of the Sr$_2^+$ molecular ion are available. Therefore, our estimation of possible error margins for excited electronic states can only be based on the ability of our approach to reproduce atomic and ionic properties as described in Sec.~\ref{sec:theory}, the convergence of results for the two lowest states presented above, and  the reproduction of exited atomic limits in molecular calculations.

The obtained excitation energies at atomic dissociation limits with the MRCISD and RCCSD(T) methods together with their experimental counterparts are presented in Table~\ref{tab:thresholds}. An analysis of these data shows that the excitation energies from the ground state to all but one excited states are reproduced by the MRCISD and RCCSD(T) methods within 5\%. The notable exception is the excitation of the Sr$_2^+$ ion to the excited states corresponding to the Sr$^+(^2D)$+Sr$(^1S)$ dissociation limit. A possible explanation of this behavior of both theoretical methods is the unsaturation of the employed basis set with $d$-type orbitals, combined with an extreme proximity of the lower-lying asymptote, which should be separated from the considered limit by 40$\,$cm$^{-1}$ only. As a result, the absolute error of this excitation energy is equal to around 2000$\,$cm$^{-1}$ for MRCISD and 1100$\,$cm$^{-1}$ for RCCSD(T), that corresponds to 14\% and 7.5\% relative errors, respectively.

Based on the above analysis and our previous experience with \textit{ab initio} calculations for similar systems, we estimate the uncertainty of the calculated interactions energies to be around 5\% for the low-lying deeply bound electronic states unaffected by avoided-crossings, whereas for high-lying states disturbed by avoided-crossings and high density of states  to be in the range of 10\% to 20\%, or even larger for weakly bound highly exited states.

It is also worth to mention, that to obtain presented converged results and smooth potential energy curves a special numerical treatment, optimized separately for all electronic symmetries has been needed including restarting calculations from different geometries and configurations, and properly adjusting  numerical thresholds.

Summarizing this part, we have shown that the alkaline-earth-metal dimers, including Sr$_2^+$ may pose a considerable challenge even for the state-of-the-art \textit{ab initio} theoretical methods. Among the possible reasons, one can mention a complicated multireference electronic structure of these species, which exhibit closely lying excited electronic states with strong radial non-adiabatic couplings. Therefore, the theoretical description of some states of the Sr$_2^+$ molecular ion may suffer from a decreased accuracy. Nonetheless, the presented results contribute significantly to the yet unknown energetic structure of this interesting molecular ion. Additionally, a demanding character of the Sr$_2^+$ ion can be treated as an opportunity to benchmark a range of theoretical methods, which are often considered as methods of choice for similar systems. A comparison of their performance in the description of the ground and some excited states has been an important aspect of our study.

\subsection{Electric dipole transition moments}

\begin{figure*}[tb]
\begin{center}
\includegraphics[width=\columnwidth]{fig8a}
\includegraphics[width=\columnwidth]{fig8b}
\includegraphics[width=\columnwidth]{fig8c}
\includegraphics[width=\columnwidth]{fig8d}
\end{center}
\caption{Franck-Condon factors between vibrational levels of the $X^1\Sigma^+_g$ ground electronic states of the $^{88}$Sr$_2$ molecule and vibrational levels of (a),(b) the $X^2\Sigma^+_u$ and (c),(d) the $1^2\Sigma^+_g$ electronic states of the $^{88}$Sr$_2^+$ molecular ion  as a function of the vibrational quantum numbers (a),(c) and vibrational energies (b),(d).}
\label{fig:FCF}
\end{figure*}

The electric dipole transition moments, necessary for a full characterization of the molecular spectra, are calculated within the MRCISD method and are presented in Fig.~\ref{fig:dip} for transitions from the $X^2\Sigma_u^+$ and $1^2\Sigma_g^+$ electronic states to all possible excited states.
The strongest transition moments reach around 9$\,$debye, e.g.~from the equilibrium interatomic distance in the $X^2\Sigma^+_u$ ground state to the $3^2\Sigma^+_g$ state as well as to the $4^2\Pi_g$ state. Some transition moments are clearly smaller for the equilibrium interatomic distance of the ground state Sr$_2^+$, suggesting that the corresponding line intensities in the spectra will be weaker. Depending on the electronic configuration of atomic limits, transitions may be dipole-allowed or dipole-forbidden for large interatomic distances. The electric dipole transition moment curves exhibit high variability and strong dependence on the interatomic distance at shorter ranges due interaction-induced modifications of the underlying electronic structure. Additionally, the complexity of the electric dipole transition moment curves may be attributed to the distortion of the excited states by many avoided crossings.

\subsection{Franck-Condon factors}

Cold molecular ions can be formed by photoassociation of laser-cooled Sr$^+$ ions immersed into an ultracold gas of Sr atoms or by ionization of ultracold Sr$_2$ molecules. Following recent successes in the production of ultracold gases of Sr$_2$ molecules, we calculate Franck-Condon factors governing the photoionization of ground-state $^{88}$Sr$_2$ molecules into the two lowest electronic states of $^{88}$Sr$_2^+$ molecular ions, while the calculation of the photoionization cross sections is out of the scope of this paper. The Schr\"odinger equation for nuclear motion is solved as implemented and described in Ref.~\cite{TomzaPRA15a}.

Fig.~\ref{fig:FCF} presents calculated Franck-Condon factors (FCFs) between vibrational levels of the $X^1\Sigma^+_g$ ground electronic state of the $^{88}$Sr$_2$ molecule and vibrational levels of the $X^2\Sigma^+_u$ and $1^2\Sigma^+_g$ electronic states of the $^{88}$Sr$_2^+$ molecular ion as a function of the vibrational quantum numbers and vibrational energies. The $X^1\Sigma^+_g$ ground state of the $^{88}$Sr$_2$ molecule supports 63 rotationless vibrational levels~\cite{SteinPRA08}, while the $X^2\Sigma^+_u$ and $1^2\Sigma^+_g$ electronic states of the $^{88}$Sr$_2^+$ molecular ion supports 263 and 116 vibrational levels, respectively. Because the ground states of Sr$_2$ and Sr$_2^+$ have similar equilibrium bond lengths, FCFs for the $X^1\Sigma^+_g$-$X^2\Sigma^+_u$ transition are regular and have a pronounced diagonal band, with the largest value of 0.38 for $v''_{X^1\Sigma^+_g}=0\to v'_{X^2\Sigma^+_u}=5$ overlap. The asymmetry visible in Fig.~\ref{fig:FCF}(a),(b) results from a significant difference in well depths of the neutral and ionic grounds states. Deeply bound Sr$_2^+$ molecular ions in the $X^2\Sigma^+_u$ state can be obtained by the ionization of deeply bound Sr$_2$ molecules, however noticeable probability can be expected for the ionization of weakly bound molecules, too. The ionization of weakly bound Sr$_2$ molecules into weakly bound molecular ions can also be expected. FCFs for the $X^1\Sigma^+_g$-$1^2\Sigma^+_g$ transition in Fig.~\ref{fig:FCF}(c),(d) are less regular due to the double well structure of the $1^2\Sigma^+_g$ state. Levels of the short-range well of the $1^2\Sigma^+_g$ state have a relatively good overlap with levels of the $X^1\Sigma^+_g$ state with the largest value of 0.62 for $v''_{X^1\Sigma^+_g}=21\to v'_{1^2\Sigma^+_g}=14$ transition. The vertical lines visible in Fig.~\ref{fig:FCF}(c),(d) are related with the FCFs to levels localized in the long-range well of the $1^2\Sigma^+_g$ state, which do not have noticeable overlap with deeply bound levels of the $X^1\Sigma^+_g$ state. Deeply bound Sr$_2^+$ molecular ions in the $1^2\Sigma^+_g$ state can be obtained by the ionization of deeply bound Sr$_2$ molecules, only. Interestingly, Sr$_2^+$ molecular ions localized in the long-range well of the $1^2\Sigma^+_g$ state can be produced by the ionization of weakly bound Sr$_2$ molecules.

\section{Summary and conclusions}
\label{sec:summary}

Cold molecular ions consisting of alkali and alkaline-earth atoms are amongst the most promising systems for interesting applications ranging from quantum simulations and precision measurements to controlled chemical reactions. In comparison to neutral molecules, molecular ions offer better preparation, trapping and detection possibilities. At the same time, while neutral alkaline-earth dimers, including several recent works on Sr$_2$, have already been intensively studied, the lack of spectroscopic data for ionic alkaline-earth dimers may limit possible experimental efforts.

In the present work, therefore, we have calculated and characterized the ground and 41~excited electronic states of the Sr$_2^+$ molecular ion and electric dipole transition moments between them. We have employed a range of \textit{ab initio} quantum-chemical methods including the coupled cluster and configuration interaction ones with small-core relativistic energy-consistent pseudopotentials and large Gaussian basis set. While low-lying potential energy curves show a smooth behavior with well defined minima, the higher-lying states display perturbations, mostly in the form of avoided crossings. We have characterized and benchmarked two lowest electronic states of the Sr$_2^+$ molecular ion with a range of computational techniques and we have observed that the Sr$_2^+$ molecular ion despite of its apparently simple structure with three valence electrons only is a challenging system even for state-of-the-art \textit{ab initio} theoretical methods due to its multireference nature and high density of excited electronic states.  Finally, we have calculated and analyzed Franck-Condon factors governing the photoionization of ground-state Sr$_2$ molecules into the two lowest electronic states of Sr$_2^+$ molecular ions. We have shown that the formation of both weakly and deeply bound Sr$_2^+$ molecular ions using photoionization of weakly and deeply bound Sr$_2$ molecules should be feasible.

As no experimental data is available for the Sr$_2^+$ molecular ion excitation energies, our results can be useful for guiding future spectroscopic measurements as well as for the formation of cold Sr$_2^+$ dimers by means of photoionization of neutral Sr$_2$ molecules from molecular beam or ultracold gas. In the future, we plan to use the calculated spectrum of the Sr$_2^+$ molecular ion in theoretical studies of an atomic ion immersed in ultracold gas of Sr$_2$ molecules where the charge transfer process is possible.
\

\section*{Supplementary Material}
Supplementary materials contain all potential energy curves and Franck-Condon factors in numerical form as plotted in all figures presented in the manuscript.

\begin{acknowledgments}
Financial support from the National Science Centre Poland (2015/19/D/ST4/02173 and 2016/23/B/ST4/03231) is gratefully acknowledged.
The computational part of this research has been partially supported by the PL-Grid Infrastructure.
\end{acknowledgments}


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